1. bookVolume 6 (2021): Issue 1 (January 2021)
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Interval Sheffer Stroke Basic Algebras and Yang-Baxter Equation

Published Online: 31 Dec 2020
Page range: 347 - 358
Received: 13 Jun 2019
Accepted: 03 Sep 2019
Journal Details
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Journal
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01 Jan 2016
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2 times per year
Languages
English
Abstract

In this study, we give basic definitions and notions about Sheffer stroke operation and Sheffer stroke basic algebra. After presenting Sheffer stroke basic algebra on a given interval, named interval Sheffer stroke basic algebra, we give some features of an interval Sheffer stroke basic algebra. Then we investigate solutions to the set-theoretical Yang-Baxter equation in this algebraic structure by using its features.

Keywords

MSC 2010

Introduction

Since any formulae or axiom in a Boolean algebra can be stated by means of Sheffer stroke operation that was preliminarily given by H.M. Sheffer [11], it is important for many researchers applying the operation to the algebraic structures such as Sheffer stroke basic algebras [15], interval Sheffer stroke basic algebras [16], Sheffer stroke NMV-algebras [7] and implication algebras [1]. Because an interval Sheffer stroke basic algebra have both properties of Sheffer stroke operation [5] and interval basic algebra ([4], [6]), it makes easier to study on an interval Sheffer stroke basic algebra which is a generalization of Sheffer stroke basic algebra.

In many different parts of science, technology and industry, many scientists wish to apply the Yang-Baxter equation that was initially used in theoretical physics [21] and statical mechanics ([2,3], [22]). In addition to the use of the equation in various scientific fields such as quantum groups, quantum mechanics, quantum computing, knot theory, integrable systems, non-commutative geometry, C*-algebras, etc. ([13, 14] and [8,9,10]), there exist many mathematicians who want to investigate set-theoretical solutions to this equation in pure mathematics. Hence, we wish to find solutions to the set-theoretical Yang-Baxter equation in an interval Sheffer stroke basic algebra.

In recent years, Oner and Katican have constructed set-theoretical solutions to the Yang-Baxter equation by using Wajsberg algebras [17] and BL-algebras [18]. Besides, Yang-Baxter equation on MTL-algebras are given by Oner and Kalkan [19]. Also, Oner et al. have presented solutions to the set-theoretical Yang-Baxter equation in MV-algebras [20]

It is first given fundamental definitions and notions about Sheffer stroke operation and Sheffer stroke basic algebra. After presenting a definition of Sheffer stroke basic algebra on a given interval, called interval Sheffer stroke basic algebra, it is given several properties of interval Sheffer stroke basic algebra. Then it is searched solutions to the set-theoretical Yang-Baxter equation on this by using these properties.

Preliminaries

It is given the following fundamental notions.

Definition 1

[5] Let 𝒰 = (U;|) be a structure. The binary operation | is called a Sheffer stroke operation if it satisfies the following conditions:

(S1) u|v = v|u,

(S2) (u|u)|(u|v) = u,

(S3) u|((v|w)|(v|w)) = ((u|v)|(u|v))|w,

(S4) (u|((u|u)|(v|v)))|(u|((u|u)|(v|v))) = u.

Oner and Senturk introduced the Sheffer Stroke basic algebra which is a basic algebra with only the Sheffer Stroke operation [15].

Definition 2

[15] An algebra 𝒰 = (U;|) of type (2) is called a Sheffer stroke basic algebra if it satisfies the following identities:

(SH1) (u|(u|u))|(u|u) = u,

(SH2) (u|(v|v))|(v|v) = (v|(u|u))|(u|u),

(SH3) (((u|(v|v))|(v|v))|(w|w))|((u|(w|w))|(u|(w|w))) = u|(u|u).

Lemma 1

[15] Let 𝒰 = (U;|) be a Sheffer Stroke basic algebra. Then there exists an algebraic constant element 1 ∈ U and 𝒰 = (U,|) provides the following identities:

(i) u|(u|u) = 1,

(ii) u|(1|1) = 1,

(iii) 1|(u|u) = u,

(iv) ((u|(v|v))|(v|v))|(v|v) = u|(v|v).

Lemma 2

[15] Let 𝒰 = (U;|) be a Sheffer Stroke basic algebra. Then the binary relation ≤, called an induced order of U, is defined by u≤v if and only if u|(v|v)=1. u \le v\; \textit{if and only if}\;u|(v|v) = 1.

Then the relation ≤ is a partial order on U such that 0 is the least element of U and 1 is the greatest element of U.

u ∨ v = (u|(v|v))|(v|v) is given in [15].

Lemma 3

[16] Let 𝒰 = (U;|) be a Sheffer Stroke basic algebra, and ≤ be the induced order of U. Then (U, ≤) is a lattice where u∨v=(u|(v|v))|(v|v) u \vee v = (u|(v|v))|(v|v) and u∧v=((u|u)∨(v|v))|((u|u)∨(v|v)). u \wedge v = ((u|u) \vee (v|v))|((u|u) \vee (v|v)).

The Interval Sheffer Stroke Basic Algebras

In this section, after presenting some definitions and notions about interval Sheffer stroke basic algebras, we give new features about interval Sheffer stroke basic algebras.

Theorem 4

[16] Let 𝒰 = (U;|) be a Sheffer Stroke basic algebra, and ≤ be the induced order of U and m, n ∈ U such that m ≤ n We define an operation | m n |_m^n on the interval [m, n] such that u | m n v : = ( n | ( u | u ) ) | ( ( ( n | ( v | v ) ) | ( m | m ) ) | ( ( n | ( v | v ) ) | ( m | m ) ) ) u|_m^nv: = (n|(u|u))|(((n|(v|v))|(m|m))|((n|(v|v))|(m|m))) for all u, v ∈ [m, n]. Then U ( m , n ) = ( [ m , n ] ; | m n ) {\cal U}(m,n) = ([m,n];|_m^n) is a Sheffer Stroke basic algebra.

By Theorem 4, the Sheffer Stroke basic algebra U ( m , n ) = ( [ m , n ] ; | m n ) {\cal U}(m,n) = ([m,n];|_m^n) , named an interval Sheffer Stroke basic algebra, satisfies (SH1) − (SH3).

Lemma 5

Let ( [ m , n ] ; | m n ) ([m,n];|_m^n) be an interval Sheffer Stroke basic algebra. Then it satisfies the following features:

u | m n ( u | m n u ) = n u|_m^n(u|_m^nu) = n

x | m n ( n | m n n ) = n x|_m^n(n|_m^nn) = n

n | m n u = u | m n u n|_m^nu = u|_m^nu

( ( u | m n ( v | m n v ) ) | m n ( v | m n v ) ) | m n ( v | m n v ) = u | m n ( v | m n v ) ((u|_m^n(v|_m^nv))|_m^n(v|_m^nv))|_m^n(v|_m^nv) = u|_m^n(v|_m^nv)

Proof

It follows from Definition 2, Theorem 4, Definition 1 and Lemma 3.

Lemma 6

Let ( [ m , n ] ; | m n ) ([m,n];|_m^n) be an interval Sheffer Stroke basic algebra. Then the binary relation ≾, called an induced order of [m, n], is defined by u ≾ v   if   and   only   if   u | m n ( v | m n v ) = n . u\precsim v\ \textit{if and only if}\ u|^{n}_{m}(v|^{n}_{m}v)=n. Then the relation ≾ is a partial order on [m, n] such that m = n | m n n m = n|_m^nn is the least element of [m, n] and n = m | m n m n = m|_m^nm is the greatest element of [m, n].

Proof

We obtain from Theorem 4, Definition 1 and Definition 2.

Lemma 7

Let ( [ m , n ] ; | m n ) ([m,n];|_m^n) be an interval Sheffer Stroke basic algebra, and ≾ be the induced order of [m, n]. Then ([m, n], ≾) is a lattice where u ∨ m n v = ( u | m n ( v | m n v ) ) | m n ( v | m n v ) u \vee _m^nv = (u|_m^n(v|_m^nv))|_m^n(v|_m^nv) and u ∧ m n v = ( ( u | m n u ) ∨ m n ( v | m n v ) ) | m n ( ( u | m n u ) ∨ m n ( v | m n v ) ) . u \wedge _m^nv = ((u|_m^nu) \vee _m^n(v|_m^nv))|_m^n((u|_m^nu) \vee _m^n(v|_m^nv)).

Proof

It is proved by Theorem 4, Definition 1, Lemma 6 and Definition 2.

Solutions to the set-theoretical Yang-Baxter Equation by interval Sheffer stroke basic algebras

In this section, we find solutions to the set-theoretical Yang-Baxter equation in interval Sheffer stroke basic algebras. Let V be a vector space over a field F. We denote by τ : V ⊗ V → V ⊗ V the twist map defined by τ(v ⊗ w) = w ⊗ v and by I : V → V the identity map over the space V; for a F-linear map R : V ⊗ V → V ⊗ V, let R12 = R ⊗ I, R23 = I ⊗ R, and R13 = (I ⊗ τ)(R ⊗ I)(τ ⊗ I).

Definition 3

[8] A Yang-Baxter operator is an invertible F-linear map R : V ⊗ V → V ⊗ V, and it satisfies the braid condition (also called the Yang-Baxter equation): R 12 ∘ R 23 ∘ R 12 = R 23 ∘ R 12 ∘ R 23 . {R^{12}} \circ {R^{23}} \circ {R^{12}} = {R^{23}} \circ {R^{12}} \circ {R^{23}}. If R satisfies Equation (1), then both R ○ τ and τ ○ R satisfy the quantum Yang-Baxter equation: R 12 ∘ R 13 ∘ R 23 = R 23 ∘ R 13 ∘ R 12 .                                                                 ( QYBE ) {R^{12}} \circ {R^{13}} \circ {R^{23}} = {R^{23}} \circ {R^{13}} \circ {R^{12}}.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;({\textit QYBE})

The following definition allows us to form a connection between the set-theoretical Yang-Baxter equation and interval Sheffer stroke basic algebras.

Definition 4

[8] Let X be a set and S : U2 → U2, S ( o 1 , o 2 ) = ( o 1 ′ , o 2 ′ ) S({o_1},{o_2}) = ({o_1^\prime},{o_2^\prime}) be a map. The map S is a solution to the set-theoretical Yang-Baxter equation if it satisfies the following identity: S 12 ∘ S 23 ∘ S 12 = S 23 ∘ S 12 ∘ S 23 , {S^{12}} \circ {S^{23}} \circ {S^{12}} = {S^{23}} \circ {S^{12}} \circ {S^{23}}, where S 12 : U 3 → U 3 ,           S 12 ( o 1 , o 2 , o 3 ) = ( o 1 ′ , o 2 ′ , o 3 ) , S 23 : U 3 → U 3 ,           S 23 ( o 1 , o 2 , o 3 ) = ( o 1 , o 2 ′ , o 3 ′ ) , S 13 : U 3 → U 3 ,           S 13 ( o 1 , o 2 , o 3 ) = ( o 1 ′ , o 2 , o 3 ′ ) . \matrix{ {{S^{12}}:{U^3} \to {U^3},\;\;\;\;\;{S^{12}}({o_1},{o_2},{o_3}) = ({o_1^\prime},{o_2^\prime},{o_3}),} \cr {{S^{23}}:{U^3} \to {U^3},\;\;\;\;\;{S^{23}}({o_1},{o_2},{o_3}) = ({o_1},{o_2^\prime},{o_3^\prime}),} \cr {{S^{13}}:{U^3} \to {U^3},\;\;\;\;\;{S^{13}}({o_1},{o_2},{o_3}) = ({o_1^\prime},{o_2},{o_3^\prime}).} \cr }

Now, we find solutions to the set-theoretical Yang-Baxter equation by using interval Sheffer stroke basic algebras.

Theorem 8

Let ( [ m , n ] ; | m n ) ([m,n];|_m^n) be an interval Sheffer stroke basic algebra. Then S ( u , v ) = ( v | m n v , u | m n u ) S(u,v) = (v|_m^nv,u|_m^nu) is a solution to the set-theoretical Yang-Baxter equation.

Proof

S12 and S23 are defined in the following forms: S 12 ( u , v , w ) = ( v | m n v , u | m n u , w ) ,           S 23 ( u , v , w ) = ( u , w | m n w , v | m n v ) . {S^{12}}(u,v,w) = (v|_m^nv,u|_m^nu,w),\;\;\;\;\;{S^{23}}(u,v,w) = (u,w|_m^nw,v|_m^nv). For all u,v,w ∈ [m,n], we get ( S 12 ∘ S 23 ∘ S 12 ) ( u , v , w ) = ( S 12 ∘ S 23 ) ( S 12 ( u , v , w ) ) = ( S 12 ∘ S 23 ) ( v | m n v , u | m n u , w ) = S 12 ( S 23 ( v | m n v , u | m n u , w ) ) = S 12 ( v | m n v , w | m n w , ( u | m n u ) | m n ( u | m n u ) ) = ( ( w | m n w ) | m n ( w | m n w ) , ( v | m n v ) | m n ( v | m n v ) , ( u | m n u ) | m n ( u | m n u ) ) = S 23 ( ( w | m n w ) | m n ( w | m n w ) , u | m n u , v | m n v ) = S 23 ( S 12 ( u , w | m n w , v | m n v ) = ( S 23 ∘ S 12 ) ( u , w | m n w , v | m n v ) = ( S 23 ∘ S 12 ) ( S 23 ( u , v , w ) ) = ( S 23 ∘ S 12 ∘ S 23 ) ( u , v , w ) \matrix{ {({S^{12}} \circ {S^{23}} \circ {S^{12}})(u,v,w)} \hfill & = \hfill & {({S^{12}} \circ {S^{23}})({S^{12}}(u,v,w))} \hfill \cr {} \hfill & = \hfill & {({S^{12}} \circ {S^{23}})(v|_m^nv,u|_m^nu,w)} \hfill \cr {} \hfill & = \hfill & {{S^{12}}({S^{23}}(v|_m^nv,u|_m^nu,w))} \hfill \cr {} \hfill & = \hfill & {{S^{12}}(v|_m^nv,w|_m^nw,(u|_m^nu)|_m^n(u|_m^nu))} \hfill \cr {} \hfill & = \hfill & {((w|_m^nw)|_m^n(w|_m^nw),(v|_m^nv)|_m^n(v|_m^nv),(u|_m^nu)|_m^n(u|_m^nu))} \hfill \cr {} \hfill & = \hfill & {{S^{23}}((w|_m^nw)|_m^n(w|_m^nw),u|_m^nu,v|_m^nv)} \hfill \cr {} \hfill & = \hfill & {{S^{23}}({S^{12}}(u,w|_m^nw,v|_m^nv)} \hfill \cr {} \hfill & = \hfill & {({S^{23}} \circ {S^{12}})(u,w|_m^nw,v|_m^nv)} \hfill \cr {} \hfill & = \hfill & {({S^{23}} \circ {S^{12}})({S^{23}}(u,v,w))} \hfill \cr {} \hfill & = \hfill & {({S^{23}} \circ {S^{12}} \circ {S^{23}})(u,v,w)} \hfill \cr } Then, S ( u , v ) = ( v | m n v , u | m n u ) S(u,v) = (v|_m^nv,u|_m^nu) is a solution to the set-theoretical Yang-Baxter equation in the interval Sheffer stroke basic algebra.

Theorem 9

Let ( [ m , n ] ; | m n ) ([m,n];|_m^n) be an interval Sheffer stroke basic algebra. Then S ( u , v ) = ( ( u | m n u ) | m n ( v | m n v ) , m ) S(u,v) = ((u|_m^nu)|_m^n(v|_m^nv),m) is a solution to the set-theoretical Yang-Baxter equation.

Proof

S12 and S23 are defined in the following forms: S 12 ( u , v , w ) = ( ( u | m n u ) | m n ( v | m n v ) , m , w ) ,           S 23 ( u , v , w ) = ( u , ( v | m n v ) | m n ( w | m n w ) , m ) . {S^{12}}(u,v,w) = ((u|_m^nu)|_m^n(v|_m^nv),m,w),\;\;\;\;\;{S^{23}}(u,v,w) = (u,(v|_m^nv)|_m^n(w|_m^nw),m). For all u,v,w ∈ [m,n], we have ( S 12 ∘ S 23 ∘ S 12 ) ( u , v , w ) = ( S 12 ∘ S 23 ) ( S 12 ( u , v , w ) ) = ( S 12 ∘ S 23 ) ( ( u | m n u ) | m n ( v | m n v ) , m , w ) = S 12 ( S 23 ( ( u | m n u ) | m n ( v | m n v ) , m , w ) ) = S 12 ( ( u | m n u ) | m n ( v | m n v ) , ( m | m n m ) | m n ( w | m n w ) , m ) = S 12 ( ( u | m n u ) | m n ( v | m n v ) , n | m n ( w | m n w ) , m ) = S 12 ( ( u | m n u ) | m n ( v | m n v ) , w , m )                                   ( Lemma   5 ( 3 )   and   ( S 2 ) ) = ( ( ( ( u | m n u ) | m n ( v | m n v ) ) | m n ( ( u | m n u ) | m n ( v | m n v ) ) ) | m n ( w | m n w ) , m , m ) = S 23 ( ( u | m n u ) | m n ( ( ( v | m n v ) | m n ( w | m n w ) ) | m n ( ( v | m n v ) | m n ( w | m n w ) ) ) , m , m )                                     ( ( S 3 )   and   ( S 2 ) ) = S 23 ( S 12 ( u , ( v | m n v ) | m n ( w | m n w ) , m ) ) = ( S 23 ∘ S 12 ) ( u , ( v | m n v ) | m n ( w | m n w ) , m ) = ( S 23 ∘ S 12 ∘ S 23 ) ( u , v , w ) = ( S 23 ∘ S 12 ) ( S 23 ( u , v , w ) ) \matrix{ {({S^{12}} \circ {S^{23}} \circ {S^{12}})(u,v,w)} \hfill & = \hfill & {({S^{12}} \circ {S^{23}})({S^{12}}(u,v,w))} \hfill \cr {} \hfill & = \hfill & {({S^{12}} \circ {S^{23}})((u|_m^nu)|_m^n(v|_m^nv),m,w)} \hfill \cr {} \hfill & = \hfill & {{S^{12}}({S^{23}}((u|_m^nu)|_m^n(v|_m^nv),m,w))} \hfill \cr {} \hfill & = \hfill & {{S^{12}}((u|_m^nu)|_m^n(v|_m^nv),(m|_m^nm)|_m^n(w|_m^nw),m)} \hfill \cr {} \hfill & = \hfill & {{S^{12}}((u|_m^nu)|_m^n(v|_m^nv),n|_m^n(w|_m^nw),m)} \hfill \cr {} \hfill & = \hfill & {{S^{12}}((u|_m^nu)|_m^n(v|_m^nv),w,m)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(Lemma\;5(3)\;and\;(S2))} \hfill \cr {} \hfill & = \hfill & {((((u|_m^nu)|_m^n(v|_m^nv))|_m^n((u|_m^nu)} \hfill \cr {} \hfill & {} \hfill & {|_m^n(v|_m^nv)))|_m^n(w|_m^nw),m,m)} \hfill \cr {} \hfill & = \hfill & {{S^{23}}((u|_m^nu)|_m^n(((v|_m^nv)|_m^n(w|_m^nw))|_m^n} \hfill \cr {} \hfill & {} \hfill & {((v|_m^nv)|_m^n(w|_m^nw))),m,m)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;((S3)\;and\;(S2))} \hfill \cr {} \hfill & = \hfill & {{S^{23}}({S^{12}}(u,(v|_m^nv)|_m^n(w|_m^nw),m))} \hfill \cr {} \hfill & = \hfill & {({S^{23}} \circ {S^{12}})(u,(v|_m^nv)|_m^n(w|_m^nw),m)} \hfill \cr {} \hfill & = \hfill & {({S^{23}} \circ {S^{12}} \circ {S^{23}})(u,v,w)} \hfill \cr {} \hfill & = \hfill & {({S^{23}} \circ {S^{12}})({S^{23}}(u,v,w))} \hfill \cr } Then, S ( u , v ) = ( ( u | m n u ) | m n ( v | m n v ) , m ) S(u,v) = ((u|_m^nu)|_m^n(v|_m^nv),m) is a solution to the set-theoretical Yang-Baxter equation in the interval Sheffer stroke basic algebra.

Theorem 10

Let ( [ m , n ] ; | m n ) ([m,n];|_m^n) be an interval Sheffer stroke basic algebra. Then S ( u , u ) = ( ( u | m n ( v | m n v ) ) | m n ( v | m n v ) , v ) S(u,u) = ((u|_m^n(v|_m^nv))|_m^n(v|_m^nv),v) is a solution to the set-theoretical Yang-Baxter equation.

Proof

S12 and S23 are defined in the following forms: S 12 ( u , v , w ) = ( ( u | m n ( v | m n v ) ) | m n ( v | m n v ) , v , w ) ,           S 23 ( u , v , w ) = ( u , ( v | m n ( w | m n w ) ) | m n ( w | m n w ) , w ) . {S^{12}}(u,v,w) = ((u|_m^n(v|_m^nv))|_m^n(v|_m^nv),v,w),\;\;\;\;\;{S^{23}}(u,v,w) = (u,(v|_m^n(w|_m^nw))|_m^n(w|_m^nw),w). For all u,v,w ∈ [m, n], we obtain ( S 12 ∘ S 23 ∘ S 12 ) ( u , v , w ) = ( S 12 ∘ S 23 ) ( S 12 ( u , v , w ) ) = ( S 12 ∘ S 23 ) ( ( u | m n ( v | m n v ) ) | m n ( v | m n v ) , v , w ) = S 12 ( S 23 ( ( u | m n ( v | m n v ) ) | m n ( v | m n v ) , v , w ) ) = S 12 ( ( u | m n ( v | m n v ) ) | m n ( v | m n v ) , ( v | m n ( w | m n w ) | m n ( w | m n w ) , w ) = ( ( ( ( u | m n ( v | m n v ) ) | m n ( v | m n v ) ) | m n ( ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) ) ) | m n ( ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) ) , ( v | m n ( w | m n w ) ) | m n ( w | m n w ) , w ) = ( ( ( ( u | m n ( v | m n v ) ) | m n ( v | m n v ) ) | m n ( ( ( w | m n ( v | m n v ) ) | m n ( v | m n v ) ) | m n ( ( w | m n ( v | m n v ) ) | m n ( v | m n v ) ) ) ) | m n ( ( ( w | m n ( v | m n v ) ) | m n ( v | m n v ) ) | m n ( ( w | m n ( v | m n v ) ) | m n ( v | m n v ) ) ) , ( v | m n ( w | m n w ) ) | m n ( w | m n w ) , w )                                                                           ( SH 2 ) = ( ( ( ( ( ( u | m n ( v | m n v ) ) | m n ( v | m n v ) ) | m n ( v | m n v ) ) | m n ( ( ( u | m n ( v | m n v ) ) | m n ( v | m n v ) ) | m n ( v | m n v ) ) ) | m n ( w | m n ( v | m n v ) ) ) | m n ( ( ( w | m n ( v | m n v ) ) | m n ( v | m n v ) ) | m n ( ( w | m n ( v | m n ) ) | m n ( v | m n v ) ) ) , ( v | m n ( w | m n w ) ) | m n ( w | m n w ) , w )                                                                 ( ( S 1 )   and   ( S 3 ) ) = ( ( ( ( u | m n ( v | m n v ) ) | m n ( u | m n ( v | m n v ) ) ) | m n ( w | m n ( v | m n v ) ) ) | m n ( ( ( w | m n ( v | m n v ) ) | m n ( v | m n v ) ) | m n ( ( w | m n ( v | m n v ) ) | m n ( v | m n v ) ) ) , ( v | m n ( w | m n w ) ) | m n ( w | m n w ) , w )                                                                                                       ( Lemma   5 ( 4 ) ) = ( ( u | m n ( ( ( w | m n ( v | m n v ) ) | m n ( v | m n v ) ) | m n ( ( w | m n ( v | m n v ) ) | m n ( v | m n v ) ) ) ) | m n ( ( ( w | m n ( v | m n v ) ) | m n ( v | m n v ) ) | m n ( ( w | m n ( v | m n v ) ) | m n ( v | m n v ) ) ) , ( v | m n ( w | m n w ) ) | m n ( w | m n w ) , w )                                                                                           ( ( S 3 )   and   ( S 1 ) ) = ( ( u | m n ( ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) ) ) | m n ( ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) ) , ( v | m n ( w | m n w ) ) | m n ( w | m n w ) , w )                                                                                     ( SH 2 ) \matrix{ {({S^{12}} \circ {S^{23}} \circ {S^{12}})(u,v,w)} \hfill & = \hfill & {({S^{12}} \circ {S^{23}})({S^{12}}(u,v,w))} \hfill \cr {} \hfill & = \hfill & {({S^{12}} \circ {S^{23}})((u|_m^n(v|_m^nv))|_m^n(v|_m^nv),v,w)} \hfill \cr {} \hfill & = \hfill & {{S^{12}}({S^{23}}((u|_m^n(v|_m^nv))|_m^n(v|_m^nv),v,w))} \hfill \cr {} \hfill & = \hfill & {{S^{12}}((u|_m^n(v|_m^nv))|_m^n(v|_m^nv),(v|_m^n(w|_m^nw)|_m^n(w|_m^nw),w)} \hfill \cr {} \hfill & = \hfill & {((((u|_m^n(v|_m^nv))|_m^n(v|_m^nv))|_m^n(((v|_m^n(w|_m^nw))} \hfill \cr {} \hfill & {} \hfill & {|_m^n(w|_m^nw))|_m^n((v|_m^n(w|_m^nw))|_m^n(w|_m^nw))))|_m^n} \hfill \cr {} \hfill & {} \hfill & {(((v|_m^n(w|_m^nw))|_m^n(w|_m^nw))|_m^n((v|_m^n(w|_m^nw))} \hfill \cr {} \hfill & {} \hfill & {|_m^n(w|_m^nw))),(v|_m^n(w|_m^nw))|_m^n(w|_m^nw),w)} \hfill \cr {} \hfill & = \hfill & {((((u|_m^n(v|_m^nv))|_m^n(v|_m^nv))|_m^n(((w|_m^n(v|_m^nv))|_m^n(v|_m^nv))|_m^n} \hfill \cr {} \hfill & {} \hfill & {((w|_m^n(v|_m^nv))|_m^n(v|_m^nv))))|_m^n(((w|_m^n(v|_m^nv))|_m^n(v|_m^nv))|_m^n} \hfill \cr {} \hfill & {} \hfill & {((w|_m^n(v|_m^nv))|_m^n(v|_m^nv))),(v|_m^n(w|_m^nw))|_m^n(w|_m^nw),w)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(SH2)} \hfill \cr {} \hfill & = \hfill & {((((((u|_m^n(v|_m^nv))|_m^n(v|_m^nv))|_m^n(v|_m^nv))|_m^n(((u|_m^n(v|_m^nv))|_m^n} \hfill \cr {} \hfill & {} \hfill & {(v|_m^nv))|_m^n(v|_m^nv)))|_m^n(w|_m^n(v|_m^nv)))|_m^n(((w|_m^n(v|_m^nv))|_m^n(v|_m^n} \hfill \cr {} \hfill & {} \hfill & {v))|_m^n((w|_m^n(v|_m^n))|_m^n(v|_m^nv))),(v|_m^n(w|_m^nw))|_m^n(w|_m^nw),w)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;((S1)\;and\;(S3))} \hfill \cr {} \hfill & = \hfill & {((((u|_m^n(v|_m^nv))|_m^n(u|_m^n(v|_m^nv)))|_m^n(w|_m^n(v|_m^n} \hfill \cr {} \hfill & {} \hfill & {v)))|_m^n(((w|_m^n(v|_m^nv))|_m^n(v|_m^nv))|_m^n((w|_m^n(v|_m^n} \hfill \cr {} \hfill & {} \hfill & {v))|_m^n(v|_m^nv))),(v|_m^n(w|_m^nw))|_m^n(w|_m^nw),w)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(Lemma\;5(4))} \hfill \cr {} \hfill & = \hfill & {((u|_m^n(((w|_m^n(v|_m^nv))|_m^n(v|_m^nv))|_m^n((w|_m^n(v|_m^nv))} \hfill \cr {} \hfill & {} \hfill & {|_m^n(v|_m^nv))))|_m^n(((w|_m^n(v|_m^nv))|_m^n(v|_m^nv))|_m^n((w|_m^n} \hfill \cr {} \hfill & {} \hfill & {(v|_m^nv))|_m^n(v|_m^nv))),(v|_m^n(w|_m^nw))|_m^n(w|_m^nw),w)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;((S3)\;and\;(S1))} \hfill \cr {} \hfill & = \hfill & {((u|_m^n(((v|_m^n(w|_m^nw))|_m^n(w|_m^nw))|_m^n((v|_m^n(w|_m^nw))} \hfill \cr {} \hfill & {} \hfill & {|_m^n(w|_m^nw))))|_m^n(((v|_m^n(w|_m^nw))|_m^n(w|_m^nw))|_m^n((v|_m^n} \hfill \cr {} \hfill & {} \hfill & {(w|_m^nw))|_m^n(w|_m^nw))),(v|_m^n(w|_m^nw))|_m^n(w|_m^nw),w)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(SH2)} \hfill \cr } and ( S 23 ∘ S 12 ∘ S 23 ) ( u , v , w ) = ( S 23 ∘ S 12 ) ( S 23 ( u , v , w ) ) = ( S 23 ∘ S 12 ) ( u , ( v | m n ( w | m n w ) ) | m n ( w | m n w ) , w ) = S 23 ( S 12 ( u , ( v | m n ( w | m n w ) ) | m n ( w | m n w ) , w ) ) = S 23 ( ( u | m n ( ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) ) ) | m n ( ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) ) , ( v | m n ( w | m n w ) ) | m n ( w | m n w ) , w ) = ( ( u | m n ( ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) ) ) | m n ( ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) ) , ( ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) | m n ( w | m n w ) ) | m n ( w | m n w ) , w ) = ( ( u | m n ( ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) ) ) | m n ( ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( w | m n w ) ) ) , ( v | m n ( w | m n w ) ) | m n ( w | m n w ) , w )                                   ( Lemma   5 ( 4 ) ) \matrix{ {({S^{23}} \circ {S^{12}} \circ {S^{23}})(u,v,w)} \hfill & = \hfill & {({S^{23}} \circ {S^{12}})({S^{23}}(u,v,w))} \hfill \cr {} \hfill & = \hfill & {({S^{23}} \circ {S^{12}})(u,(v|_m^n(w|_m^nw))|_m^n(w|_m^nw),w)} \hfill \cr {} \hfill & = \hfill & {{S^{23}}({S^{12}}(u,(v|_m^n(w|_m^nw))|_m^n(w|_m^nw),w))} \hfill \cr {} \hfill & = \hfill & {{S^{23}}((u|_m^n(((v|_m^n(w|_m^nw))|_m^n(w|_m^nw))|_m^n((v|_m^n(w|_m^nw))} \hfill \cr {} \hfill & {} \hfill & {|_m^n(w|_m^nw))))|_m^n(((v|_m^n(w|_m^nw))|_m^n(w|_m^nw))|_m^n((v|_m^n} \hfill \cr {} \hfill & {} \hfill & {(w|_m^nw))|_m^n(w|_m^nw))),(v|_m^n(w|_m^nw))|_m^n(w|_m^nw),w)} \hfill \cr {} \hfill & = \hfill & {((u|_m^n(((v|_m^n(w|_m^nw))|_m^n(w|_m^nw))|_m^n((v|_m^n(w|_m^nw))|_m^n(w|_m^n} \hfill \cr {} \hfill & {} \hfill & {w))))|_m^n(((v|_m^n(w|_m^nw))|_m^n(w|_m^nw))|_m^n((v|_m^n(w|_m^nw))|_m^n(w} \hfill \cr {} \hfill & {} \hfill & {|_m^nw))),(((v|_m^n(w|_m^nw))|_m^n(w|_m^nw))|_m^n(w|_m^nw))|_m^n(w|_m^nw),w)} \hfill \cr {} \hfill & = \hfill & {((u|_m^n(((v|_m^n(w|_m^nw))|_m^n(w|_m^nw))|_m^n((v|_m^n(w|_m^nw))|_m^n} \hfill \cr {} \hfill & {} \hfill & {(w|_m^nw))))|_m^n(((v|_m^n(w|_m^nw))|_m^n(w|_m^nw))|_m^n((v|_m^n} \hfill \cr {} \hfill & {} \hfill & {(w|_m^nw))|_m^n(w|_m^nw))),(v|_m^n(w|_m^nw))|_m^n(w|_m^nw),w)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(Lemma\;5(4))} \hfill \cr } Then, S ( u , v ) = ( ( u | m n ( v | m n v ) ) | m n ( v | m n v ) , v ) S(u,v) = ((u|_m^n(v|_m^nv))|_m^n(v|_m^nv),v) is a solution to the set-theoretical Yang-Baxter equation in the interval Sheffer stroke basic algebra.

Corollary 11

Let ( [ m , n ] ; | m n ) ([m,n];|_m^n) be an interval Sheffer stroke basic algebra. Then S ( u , v ) = ( ( v | m n ( u | m n u ) ) | m n ( u | m n u ) , u ) S(u,v) = ((v|_m^n(u|_m^nu))|_m^n(u|_m^nu),u) and S ( u , v ) = ( ( u | m n ( v | m n v ) | m n ( v | m n v ) , u ) S(u,v) = ((u|_m^n(v|_m^nv)|_m^n(v|_m^nv),u) are solutions to the set-theoretical Yang-Baxter equation.

Proof

We get from (SH2) and Theorem 10.

Theorem 12

Let ( [ m , n ] ; | m n ) ([m,n];|_m^n) be an interval Sheffer stroke basic algebra. Then S ( u , v ) = ( ( u | a b ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) , u ) S(u,v) = ((u|_a^b(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))),u) is a solution to the set-theoretical Yang-Baxter equation.

Proof

S12 and S23 are defined in the following forms: S 12 ( u , v , w ) = ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) , u , w ) , S 23 ( u , v , w ) = ( u , ( v | m n ( v | m n ( w | m n w ) ) ) | m n ( v | m n ( v | m n ( w | m n w ) ) ) , v ) . \matrix{ {{S^{12}}(u,v,w) = ((u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))),u,w),} \hfill \cr {{S^{23}}(u,v,w) = (u,(v|_m^n(v|_m^n(w|_m^nw)))|_m^n(v|_m^n(v|_m^n(w|_m^nw))),v).} \hfill \cr } For all u,v,w ∈ [m,n], we obtain ( S 12 ∘ S 23 ∘ S 12 ) ( u , v , w ) = ( S 12 ∘ S 23 ) ( S 12 ( u , v , w ) ) = ( S 12 ∘ S 23 ) ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) , u , w ) = S 12 ( S 23 ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) , u , w ) ) = S 12 ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) , ( u | a b ( u | a b ( w | a b w ) ) ) | a b ( u | a b ( u | a b ( w | a b w ) ) ) , u ) = ( ( ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) ) | m n ( ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) ) | m n ( u | m n ( u | m n ( w | m n w ) ) ) ) ) | m n ( ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) ) | m n ( ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) ) | m n ( u | m n ( u | m n ( w | m n w ) ) ) ) ) , ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) , u )                                         ( S 2 ) = ( ( ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) ) | m n ( ( u | m n ( v | m n v ) ) | m n ( ( u | m n ( u | m n ( u | m n ( w | m n w ) ) ) ) | m n ( u | m n ( u | m n ( u | m n ( w | m n w ) ) ) ) ) ) ) | m n ( ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) ) | m n ( ( u | m n ( v | m n v ) ) | m n ( ( u | m n ( u | m n ( u | m n ( w | m n w ) ) ) ) | m n ( u | m n ( u | m n ( u | m n ( w | m n w ) ) ) ) ) ) ) , ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) , u )                                       ( ( S 1 )   and   ( S 3 ) ) = ( ( ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) ) | m n ( ( u | m n ( v | m n v ) ) | m n ( ( u | m n ( w | m n w ) ) | m n ( u | m n ( w | m n w ) ) ) ) ) | m n ( ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) ) | m n ( ( u | m n ( v | m n v ) ) | m n ( ( u | m n ( w | m n w ) ) | m n ( u | m n ( w | m n w ) ) ) ) ) , ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n y ) ) ) , u )                                       ( ( S 1 ) ,   ( S 3 )   and   Lemma   5   ( 4 ) ) = ( ( ( ( w | m n w ) | m n ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) ) ) | m n ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) ) ) | m n ( ( ( w | m n w ) | m n ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) ) ) | m n ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) ) ) , ( u | m n ( u | m n ( v | m n v ) ) ) | n n ( u | m n ( u | m n ( v | m n v ) ) ) , u )                         ( ( S 3 )   and   ( S 1 ) ) = ( ( ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n w ) | m n w ) | m n ( ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n w ) | m n w ) , ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | a b v ) ) ) , u )                                                         ( ( SH 2 )   and   ( S 2 ) ) = ( ( ( ( v | m n ( v | m n ( u | m n u ) ) ) | m n w ) | m n w ) | m n ( ( ( v | m n ( v | m n ( u | m n u ) ) ) | n n w ) | m n w ) , ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) , u )                                                           ( ( S 1 ) , ( S 2 )   and   ( SH 2 ) ) = ( ( ( ( w | m n w ) | m n ( ( v | m n ( v | m n ( u | m n u ) ) ) | m n ( v | m n ( v | m n ( u | m n u ) ) ) ) ) | m n ( ( v | m n ( v | m n ( u | m n u ) ) ) | m n ( v | m n ( v | m n ( u | m n u ) ) ) ) ) | m n ( ( ( w | m n w ) | m n ( ( v | m n ( v | m n ( u | m n u ) ) ) | m n ( v | m n ( v | m n ( u | m n u ) ) ) ) ) | m n ( ( v | m n ( v | m n ( u | m n u ) ) ) | m n ( v | m n ( v | m n ( u | m n u ) ) ) ) ) , ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) , u )                                                 ( ( S 2 )   and   ( SH 2 ) ) = ( ( v | m n ( ( ( v | m n ( u | m n u ) ) | m n ( ( v | m n ( u | m n u ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( v | m n ( w | m n w ) ) ) ) ) | m n ( ( v | m n ( u | m n u ) ) | m n ( ( v | m n ( u | m n u ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( v | m n ( w | m n w ) ) ) ) ) ) ) | m n ( v | m n ( ( ( v | m n ( u | m n u ) ) | m n ( ( v | m n ( u | m n u ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( v | m n ( w | m n w ) ) ) ) ) | m n ( ( v | m n ( u | m n u ) ) | m n ( ( v | m n ( u | m n u ) ) | m n ( ( v | m n ( w | n n w ) ) | m n ( v | m n ( w | m n w ) ) ) ) ) ) ) , ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) , u )                             ( ( S 1 )   and   ( S 3 ) ) = ( ( v | m n ( ( ( v | m n ( w | m n w ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( ( v | m n ( u | m n u ) ) | m n ( v | m n ( u | m n u ) ) ) ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( ( v | m n ( u | m n u ) ) | m n ( v | m n ( u | m n u ) ) ) ) ) ) ) | m n ( v | m n ( ( ( v | m n ( w | m n w ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( ( v | m n ( u | m n u ) ) | m n ( v | m n ( u | m n u ) ) ) ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( ( v | m n ( w | m n w ) ) | m n ( ( v | m n ( u | m n u ) ) | m n ( v | m n ( u | m n u ) ) ) ) ) ) ) , ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) , u )                             ( ( S 1 ) ,   ( S 2 )   and   ( SH 2 ) ) = ( ( ( ( u | m n u ) | m n ( ( v | m n ( v | m n ( w | m n w ) ) ) | m n ( v | m n ( v | m n ( w | m n w ) ) ) ) ) | m n ( ( v | m n ( v | m n ( w | m n w ) ) ) | m n ( v | m n ( v | m n ( w | m n w ) ) ) ) ) | m n ( ( ( u | m n u ) | m n ( ( v | m n ( v | m n ( w | m n w ) ) ) | m n ( v | m n ( v | m n ( w | m n w ) ) ) ) ) | m n ( ( v | m n ( v | m n ( w | m n w ) ) ) | m n ( v | m n ( v | m n ( w | m n w ) ) ) ) ) , ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) , u )                                 ( ( S 3 )   and   ( S 1 ) ) = ( ( u | m n ( u | m n ( v | m n ( v | m n ( w | m n w ) ) ) ) ) | m n ( u | m n ( u | m n ( v | m n ( v | m n ( w | m n w ) ) ) ) ) , ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) , u )                     ( ( SH 2 ) ,   ( S 2 )   and   ( S 1 ) ) \matrix{ {({S^{12}} \circ {S^{23}} \circ {S^{12}})(u,v,w)} \hfill & = \hfill & {({S^{12}} \circ {S^{23}})({S^{12}}(u,v,w))} \hfill \cr {} \hfill & = \hfill & {({S^{12}} \circ {S^{23}})((u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))),u,w)} \hfill \cr {} \hfill & = \hfill & {{S^{12}}({S^{23}}((u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))),u,w))} \hfill \cr {} \hfill & = \hfill & {{S^{12}}((u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))),} \hfill \cr {} \hfill & {} \hfill & {(u|_a^b(u|_a^b(w|_a^bw)))|_a^b(u|_a^b(u|_a^b(w|_a^bw))),u)} \hfill \cr {} \hfill & = \hfill & {((((u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))))|_m^n} \hfill \cr {} \hfill & {} \hfill & {(((u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))))|_m^n} \hfill \cr {} \hfill & {} \hfill & {(u|_m^n(u|_m^n(w|_m^nw)))))|_m^n(((u|_m^n(u|_m^n(v|_m^nv)))|_m^n} \hfill \cr {} \hfill & {} \hfill & {(u|_m^n(u|_m^n(v|_m^nv))))|_m^n(((u|_m^n(u|_m^n(v|_m^nv)))|_m^n} \hfill \cr {} \hfill & {} \hfill & {(u|_m^n(u|_m^n(v|_m^nv))))|_m^n(u|_m^n(u|_m^n(w|_m^nw))))),} \hfill \cr {} \hfill & {} \hfill & {(u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))),u)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(S2)} \hfill \cr {} \hfill & = \hfill & {((((u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))))|_m^n} \hfill \cr {} \hfill & {} \hfill & {((u|_m^n(v|_m^nv))|_m^n((u|_m^n(u|_m^n(u|_m^n(w|_m^nw))))|_m^n} \hfill \cr {} \hfill & {} \hfill & {(u|_m^n(u|_m^n(u|_m^n(w|_m^nw)))))))|_m^n(((u|_m^n(u|_m^n(v|_m^nv)))} \hfill \cr {} \hfill & {} \hfill & {|_m^n(u|_m^n(u|_m^n(v|_m^nv))))|_m^n((u|_m^n(v|_m^nv))|_m^n((u|_m^n} \hfill \cr {} \hfill & {} \hfill & {(u|_m^n(u|_m^n(w|_m^nw))))|_m^n(u|_m^n(u|_m^n(u|_m^n(w|_m^nw))))))),} \hfill \cr {} \hfill & {} \hfill & {(u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))),u)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;((S1)\;and\;(S3))} \hfill \cr {} \hfill & = \hfill & {((((u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))))|_m^n} \hfill \cr {} \hfill & {} \hfill & {((u|_m^n(v|_m^nv))|_m^n((u|_m^n(w|_m^nw))|_m^n(u|_m^n(w|_m^nw)))))} \hfill \cr {} \hfill & {} \hfill & {|_m^n(((u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))))|_m^n} \hfill \cr {} \hfill & {} \hfill & {((u|_m^n(v|_m^nv))|_m^n((u|_m^n(w|_m^nw))|_m^n(u|_m^n(w|_m^nw))))),} \hfill \cr {} \hfill & {} \hfill & {(u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^ny))),u)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;((S1),\;(S3)\;and\;Lemma\;5\;(4))} \hfill \cr {} \hfill & = \hfill & {((((w|_m^nw)|_m^n((u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^n} \hfill \cr {} \hfill & {} \hfill & {v)))))|_m^n((u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv)))))} \hfill \cr {} \hfill & {} \hfill & {|_m^n(((w|_m^nw)|_m^n((u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n} \hfill \cr {} \hfill & {} \hfill & {(v|_m^nv)))))|_m^n((u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^n} \hfill \cr {} \hfill & {} \hfill & {v))))),(u|_m^n(u|_m^n(v|_m^nv)))|_n^n(u|_m^n(u|_m^n(v|_m^nv))),u)\;\;\;\;\;\;\;\;\;((S3)\;and\;(S1))} \hfill \cr {} \hfill & = \hfill & {((((u|_m^n(u|_m^n(v|_m^nv)))|_m^nw)|_m^nw)|_m^n} \hfill \cr {} \hfill & {} \hfill & {(((u|_m^n(u|_m^n(v|_m^nv)))|_m^nw)|_m^nw),(u|_m^n} \hfill \cr {} \hfill & {} \hfill & {(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_a^bv))),u)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;((SH2)\;and\;(S2))} \hfill \cr {} \hfill & = \hfill & {((((v|_m^n(v|_m^n(u|_m^nu)))|_m^nw)|_m^nw)|_m^n} \hfill \cr {} \hfill & {} \hfill & {(((v|_m^n(v|_m^n(u|_m^nu)))|_n^nw)|_m^nw), (u|_m^n} \hfill \cr {} \hfill & {} \hfill & {(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))),u)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;((S1), (S2)\;and\;(SH2))} \hfill \cr {} \hfill & = \hfill & {((((w|_m^nw)|_m^n((v|_m^n(v|_m^n(u|_m^nu)))|_m^n(v|_m^n(v|_m^n(u|_m^n} \hfill \cr {} \hfill & {} \hfill & {u)))))|_m^n((v|_m^n(v|_m^n(u|_m^nu)))|_m^n(v|_m^n(v|_m^n(u|_m^nu)))))} \hfill \cr {} \hfill & {} \hfill & {|_m^n(((w|_m^nw)|_m^n((v|_m^n(v|_m^n(u|_m^nu)))|_m^n(v|_m^n(v|_m^n(u|_m^n} \hfill \cr {} \hfill & {} \hfill & {u)))))|_m^n((v|_m^n(v|_m^n(u|_m^nu)))|_m^n(v|_m^n(v|_m^n(u|_m^nu))))),} \hfill \cr {} \hfill & {} \hfill & {(u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))),u)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;((S2)\;and\;(SH2))} \hfill \cr {} \hfill & = \hfill & {((v|_m^n(((v|_m^n(u|_m^nu))|_m^n((v|_m^n(u|_m^nu))|_m^n((v|_m^n(w|_m^nw))|_m^n} \hfill \cr {} \hfill & {} \hfill & {(v|_m^n(w|_m^nw)))))|_m^n((v|_m^n(u|_m^nu))|_m^n((v|_m^n(u|_m^nu))|_m^n((v} \hfill \cr {} \hfill & {} \hfill & {|_m^n(w|_m^nw))|_m^n(v|_m^n(w|_m^nw)))))))|_m^n(v|_m^n(((v|_m^n(u|_m^nu))|_m^n} \hfill \cr {} \hfill & {} \hfill & {((v|_m^n(u|_m^nu))|_m^n((v|_m^n(w|_m^nw))|_m^n(v|_m^n(w|_m^nw)))))|_m^n((v} \hfill \cr {} \hfill & {} \hfill & {|_m^n(u|_m^nu))|_m^n((v|_m^n(u|_m^nu))|_m^n((v|_m^n(w|_n^nw))|_m^n(v|_m^n(w|_m^n} \hfill \cr {} \hfill & {} \hfill & {w))))))), (u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))),u)\;\;\;\;\;\;\;\;\;((S1)\;and\;(S3))} \hfill \cr {} \hfill & = \hfill & {((v|_m^n(((v|_m^n(w|_m^nw))|_m^n((v|_m^n(w|_m^nw))|_m^n((v|_m^n(u|_m^nu))|_m^n} \hfill \cr {} \hfill & {} \hfill & {(v|_m^n(u|_m^nu)))))|_m^n((v|_m^n(w|_m^nw))|_m^n((v|_m^n(w|_m^nw))|_m^n((v} \hfill \cr {} \hfill & {} \hfill & {|_m^n(u|_m^nu))|_m^n(v|_m^n(u|_m^nu)))))))|_m^n(v|_m^n(((v|_m^n(w|_m^nw))|_m^n} \hfill \cr {} \hfill & {} \hfill & {((v|_m^n(w|_m^nw))|_m^n((v|_m^n(u|_m^nu))|_m^n(v|_m^n(u|_m^nu)))))|_m^n((v} \hfill \cr {} \hfill & {} \hfill & {|_m^n(w|_m^nw))|_m^n((v|_m^n(w|_m^nw))|_m^n((v|_m^n(u|_m^nu))|_m^n(v|_m^n(u|_m^n} \hfill \cr {} \hfill & {} \hfill & {u))))))),(u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))),u)\;\;\;\;\;\;\;\;\;\;\;\;((S1),\;(S2)\;and\;(SH2))} \hfill \cr {} \hfill & = \hfill & {((((u|_m^nu)|_m^n((v|_m^n(v|_m^n(w|_m^nw)))|_m^n(v|_m^n(v|_m^n(w|_m^n} \hfill \cr {} \hfill & {} \hfill & {w)))))|_m^n((v|_m^n(v|_m^n(w|_m^nw)))|_m^n(v|_m^n(v|_m^n(w|_m^nw)))))} \hfill \cr {} \hfill & {} \hfill & {|_m^n(((u|_m^nu)|_m^n((v|_m^n(v|_m^n(w|_m^nw)))|_m^n(v|_m^n(v|_m^n(w|_m^n} \hfill \cr {} \hfill & {} \hfill & {w)))))|_m^n((v|_m^n(v|_m^n(w|_m^nw)))|_m^n(v|_m^n(v|_m^n(w|_m^n} \hfill \cr {} \hfill & {} \hfill & {w))))),(u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))),u)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;((S3)\;and\;(S1))} \hfill \cr {} \hfill & = \hfill & {((u|_m^n(u|_m^n(v|_m^n(v|_m^n(w|_m^nw)))))|_m^n(u|_m^n(u|_m^n(v|_m^n(v|_m^n} \hfill \cr {} \hfill & {} \hfill & {(w|_m^nw))))),(u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))),u)\;\;\;\;\;\;\;((SH2),\;(S2)\;and\;(S1))} \hfill \cr {} \hfill & {} \hfill & {} \hfill \cr } and ( S 23 ∘ S 12 ∘ S 23 ) ( u , v , w ) = ( S 23 ∘ S 12 ) ( S 23 ( u , v , w ) ) = ( S 23 ∘ S 12 ) ( u , ( v | m n ( v | m n ( w | m n w ) ) ) | m n ( v | m n ( v | m n ( w | m n w ) ) ) , v ) = S 23 ( S 12 ( u , ( v | m n ( v | m n ( w | m n w ) ) ) | m n ( v | m n ( v | m n ( w | m n w ) ) ) , v ) ) = S 23 ( ( u | m n ( u | m n ( v | m n ( v | m n ( w | m n w ) ) ) ) ) | m n ( u | m n ( u | m n ( v | m n ( v | m n ( w | m n w ) ) ) ) ) , u , v )                                                                             ( S 2 ) = ( ( u | m n ( u | m n ( v | m n ( v | m n ( w | m n w ) ) ) ) ) | m n ( u | m n ( u | m n ( v | m n ( v | m n ( w | m n w ) ) ) ) ) , ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) , u ) \matrix{ {({S^{23}} \circ {S^{12}} \circ {S^{23}})(u,v,w)} \hfill & = \hfill & {({S^{23}} \circ {S^{12}})({S^{23}}(u,v,w))} \hfill \cr {} \hfill & = \hfill & {({S^{23}} \circ {S^{12}})(u,(v|_m^n(v|_m^n(w|_m^nw)))|_m^n(v|_m^n(v|_m^n(w|_m^nw))),v)} \hfill \cr {} \hfill & = \hfill & {{S^{23}}({S^{12}}(u,(v|_m^n(v|_m^n(w|_m^nw)))|_m^n(v|_m^n(v|_m^n(w|_m^nw))),v))} \hfill \cr {} \hfill & = \hfill & {{S^{23}}((u|_m^n(u|_m^n(v|_m^n(v|_m^n(w|_m^nw)))))|_m^n} \hfill \cr {} \hfill & {} \hfill & {(u|_m^n(u|_m^n(v|_m^n(v|_m^n(w|_m^nw))))),u,v)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(S2)} \hfill \cr {} \hfill & = \hfill & {((u|_m^n(u|_m^n(v|_m^n(v|_m^n(w|_m^nw)))))|_m^n(u|_m^n(u|_m^n(v|_m^n(v|_m^n} \hfill \cr {} \hfill & {} \hfill & {(w|_m^nw))))),(u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))),u)} \hfill \cr } Then, S ( u , v ) = ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) , u ) S(u,v) = ((u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))),u) is a solution to the set-theoretical Yang-Baxter equation in the interval Sheffer stroke basic algebra.

Corollary 13

Let ( [ m , n ] ; | m n ) ([m,n];|_m^n) be an interval Sheffer stroke basic algebra. Then S ( u , v ) = ( ( v | m n ( v | m n ( u | m n u ) ) ) | m n ( v | m n ( v | m n ( u | m n u ) ) ) , u ) S(u,v) = ((v|_m^n(v|_m^n(u|_m^nu)))|_m^n(v|_m^n(v|_m^n(u|_m^nu))),u) and S ( u , v ) = ( ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) , v ) S(u,v) = ((u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))),v) are solutions to the set-theoretical Yang-Baxter equation.

Proof

It is proved from (S1), (S2), (SH2) and Theorem 12.

Theorem 14

Let ( [ m , n ] ; | m n ) ([m,n];|_m^n) be an interval Sheffer stroke basic algebra. Then S ( u , v ) = ( u ∨ m n v , u ∧ m n v ) S(u,v) = (u \vee _m^nv,u \wedge _m^nv) is a solution to the set-theoretical Yang-Baxter equation.

Proof

S12 and S23 are defined in the following forms: S 12 ( u , v , w ) = ( u ∨ m n v , u ∧ m n v , z ) ,       S 23 ( u , v , w ) = ( u , v ∨ m n w , v ∧ m n w ) . {S^{12}}(u,v,w) = (u \vee _m^nv,u \wedge _m^nv,z),\;\;\;\;\;{S^{23}}(u,v,w) = (u,v \vee _m^nw,v \wedge _m^nw). For all u,v,w ∈ [m,n], it follows ( S 12 ∘ S 23 ∘ S 12 ) ( u , v , w ) = ( S 12 ∘ S 23 ) ( S 12 ( u , v , w ) ) = ( S 12 ∘ S 23 ) ( u ∨ m n v , u ∧ m n v , w ) = S 12 ( S 23 ( u ∨ m n v , u ∧ m n v , w ) ) = S 12 ( u ∨ m n v , ( u ∧ m n v ) ∨ m n w , ( u ∧ m n v ) ∧ m n w ) = ( ( u ∨ m n v ) ∨ m n ( ( u ∧ m n v ) ∨ m n w ) , ( u ∨ m n v ) ∧ m n ( ( u ∧ m n v ) ∨ m n w ) , ( u ∧ m n v ) ∧ m n w ) = ( ( ( u ∨ m n v ) ∨ m n ( u ∧ m n v ) ) ∨ m n w , ( u ∨ m n v ) ∧ m n ( ( u ∨ m n w ) ∧ m n ( v ∨ m n w ) ) , u ∧ m n ( v ∧ m n w ) ) = ( ( u ∨ m n v ) ∨ m n w , ( ( u ∨ m n v ) ∧ m n ( u ∨ m n w ) ) ∧ m n ( v ∨ m n w ) , u ∧ m n ( v ∧ m n w ) ) = ( u ∨ m n ( v ∨ m n w ) , ( u ∨ m n ( v ∧ m n w ) ) ∧ m n ( v ∨ m n w ) , u ∧ m n ( v ∧ m n w ) ) \matrix{ {({S^{12}} \circ {S^{23}} \circ {S^{12}})(u,v,w)} \hfill & = \hfill & {({S^{12}} \circ {S^{23}})({S^{12}}(u,v,w))} \hfill \cr {} \hfill & = \hfill & {({S^{12}} \circ {S^{23}})(u \vee _m^nv,u \wedge _m^nv,w)} \hfill \cr {} \hfill & = \hfill & {{S^{12}}({S^{23}}(u \vee _m^nv,u \wedge _m^nv,w))} \hfill \cr {} \hfill & = \hfill & {{S^{12}}(u \vee _m^nv,(u \wedge _m^nv) \vee _m^nw,(u \wedge _m^nv) \wedge _m^nw)} \hfill \cr {} \hfill & = \hfill & {((u \vee _m^nv) \vee _m^n((u \wedge _m^nv) \vee _m^nw),(u \vee _m^nv)} \hfill \cr {} \hfill & {} \hfill & { \wedge _m^n((u \wedge _m^nv) \vee _m^nw),(u \wedge _m^nv) \wedge _m^nw)} \hfill \cr {} \hfill & = \hfill & {(((u \vee _m^nv) \vee _m^n(u \wedge _m^nv)) \vee _m^nw,(u \vee _m^nv) \wedge _m^n} \hfill \cr {} \hfill & {} \hfill & {((u \vee _m^nw) \wedge _m^n(v \vee _m^nw)),u \wedge _m^n(v \wedge _m^nw))} \hfill \cr {} \hfill & = \hfill & {((u \vee _m^nv) \vee _m^nw,((u \vee _m^nv) \wedge _m^n(u \vee _m^n} \hfill \cr {} \hfill & {} \hfill & {w)) \wedge _m^n(v \vee _m^nw),u \wedge _m^n(v \wedge _m^nw))} \hfill \cr {} \hfill & = \hfill & {(u \vee _m^n(v \vee _m^nw),(u \vee _m^n(v \wedge _m^nw))} \hfill \cr {} \hfill & {} \hfill & { \wedge _m^n(v \vee _m^nw),u \wedge _m^n(v \wedge _m^nw))} \hfill \cr {} \hfill & {} \hfill & {} \hfill \cr } and ( S 23 ∘ S 12 ∘ S 23 ) ( u , v , w ) = ( S 23 ∘ S 12 ) ( S 23 ( u , v , w ) ) = ( S 23 ∘ S 12 ) ( u , v ∨ m n w , v ∧ m n w ) = S 23 ( S 12 ( u , v ∨ m n w , v ∧ m n w ) ) = S 23 ( u ∨ m n ( v ∨ a n w ) , u ∧ m n ( v ∨ m n w ) , v ∧ m n w ) = ( u ∨ m n ( v ∨ m n w ) , ( u ∧ m n ( v ∨ m n w ) ) ∨ m n ( v ∧ m n w ) , ( u ∧ m n ( v ∨ m n w ) ) ∧ m n ( v ∧ m n w ) ) = ( u ∨ m n ( v ∨ m n w ) , ( u ∨ m n ( v ∧ m n w ) ) ∧ m n ( ( v ∨ m n w ) ∨ m n ( v ∧ m n w ) ) , u ∧ m n ( ( v ∨ m n w ) ∧ m n ( v ∧ m n w ) ) ) = ( u ∨ m n ( v ∨ m n w ) , ( u ∨ m n ( v ∧ m n w ) ) ∧ m n ( v ∨ m n w ) , u ∧ m n ( v ∧ m n w ) ) \matrix{ {({S^{23}} \circ {S^{12}} \circ {S^{23}})(u,v,w)} \hfill & = \hfill & {({S^{23}} \circ {S^{12}})({S^{23}}(u,v,w))} \hfill \cr {} \hfill & = \hfill & {({S^{23}} \circ {S^{12}})(u,v \vee _m^nw,v \wedge _m^nw)} \hfill \cr {} \hfill & = \hfill & {{S^{23}}({S^{12}}(u,v \vee _m^nw,v \wedge _m^nw))} \hfill \cr {} \hfill & = \hfill & {{S^{23}}(u \vee _m^n(v \vee _a^nw),u \wedge _m^n(v \vee _m^nw),v \wedge _m^nw)} \hfill \cr {} \hfill & = \hfill & {(u \vee _m^n(v \vee _m^nw),(u \wedge _m^n(v \vee _m^nw)) \vee _m^n(v} \hfill \cr {} \hfill & {} \hfill & { \wedge _m^nw),(u \wedge _m^n(v \vee _m^nw)) \wedge _m^n(v \wedge _m^nw))} \hfill \cr {} \hfill & = \hfill & {(u \vee _m^n(v \vee _m^nw),(u \vee _m^n(v \wedge _m^nw)) \wedge _m^n((v \vee _m^nw)} \hfill \cr {} \hfill & {} \hfill & { \vee _m^n(v \wedge _m^nw)),u \wedge _m^n((v \vee _m^nw) \wedge _m^n(v \wedge _m^nw)))} \hfill \cr {} \hfill & = \hfill & {(u \vee _m^n(v \vee _m^nw),(u \vee _m^n(v \wedge _m^nw))} \hfill \cr {} \hfill & {} \hfill & { \wedge _m^n(v \vee _m^nw),u \wedge _m^n(v \wedge _m^nw))} \hfill \cr } Then, S ( u , v ) = ( u ∨ m n v , u ∧ m n v ) S(u,v) = (u \vee _m^nv,u \wedge _m^nv) is a solution to the set-theoretical Yang-Baxter equation in the interval Sheffer stroke basic algebra.

Remark 1

S(u,v) = (u ∨ v,u ∧ v) is generally not a solution to the set-theoretical Yang-Baxter equation in a Wajsberg algebra [17] while it is a solution to the set-theoretical Yang-Baxter equation in an interval Sheffer stroke basic algebra, and also in a Boolean algebra [9].

Theorem 15

Let ( [ m , n ] ; | m n ) ([m,n];|_m^n) be an interval Sheffer stroke basic algebra. If w | m n ( ( u | m n u ) | m n ( v | m n v ) ) = ( ( u | m n w ) | m n ( v | m n w ) ) | m n ( ( u | m n w ) | m n ( v | m n w ) ) w|_m^n((u|_m^nu)|_m^n(v|_m^nv)) = ((u|_m^nw)|_m^n(v|_m^nw))|_m^n((u|_m^nw)|_m^n(v|_m^nw)) holds for all u,v,w ∈ [m,n], then S ( u , v ) = ( ( u | m n u ) | m n ( v | m n v ) , ( u | m n v ) | m n ( u | m n v ) ) S(u,v) = ((u|_m^nu)|_m^n(v|_m^nv),(u|_m^nv)|_m^n(u|_m^nv)) is a solution to the set-theoretical Yang-Baxter equation.

Proof

By Theorem 14, we know that S ( u , v ) = ( u ∨ m n v , u ∧ m n v ) S(u,v) = (u \vee _m^nv,u \wedge _m^nv) . Then it follows from Lemma 7, (S1), (S2) and (SH2) that S ( u , v ) = ( u ∨ m n v , u ∧ m n v ) = ( ( u | m n ( v | m n v ) ) | m n ( v | m n v ) , ( ( u | m n u ) ∨ m n ( v | m n v ) ) | m n ( ( u | m n u ) ∨ m n ( v | m n v ) ) ) = ( ( u | m n ( v | a b v ) ) | m n ( v | m n v ) , ( ( ( u | m n u ) | m n ( ( v | m n v ) | m n ( v | m n v ) ) ) | m n ( ( v | m n v ) | m n ( v | m n v ) ) ) | m n ( ( ( u | m n u ) | m n ( ( v | m n v ) | m n ( v | m n v ) ) ) | m n ( ( v | m n v ) | m n ( v | m n v ) ) ) ) = ( ( u | m n ( v | m n v ) ) | m n ( v | m n v ) , ( ( ( v | m n v ) | m n ( ( u | m n u ) | m n ( u | m n u ) ) ) | m n ( ( u | m n u ) | m n ( u | m n u ) ) ) | m n ( ( ( v | m n v ) | m n ( ( u | m n u ) | m n ( u | m n u ) ) ) | m n ( ( u | m n u ) | m n ( u | m n u ) ) ) ) = ( ( u | m n ( v | m n v ) ) | m n ( v | m n v ) , ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) ) \matrix{ {S(u,v) = (u \vee _m^nv,u \wedge _m^nv)} \hfill & = \hfill & {((u|_m^n(v|_m^nv))|_m^n(v|_m^nv),((u|_m^nu) \vee _m^n(v|_m^nv))|_m^n((u|_m^nu) \vee _m^n(v|_m^nv)))} \hfill \cr {} \hfill & = \hfill & {((u|_m^n(v|_a^bv))|_m^n(v|_m^nv),(((u|_m^nu)|_m^n((v|_m^nv)|_m^n(v|_m^nv)))|_m^n((v|_m^nv)|_m^n} \hfill \cr {} \hfill & {} \hfill & {(v|_m^nv)))|_m^n(((u|_m^nu)|_m^n((v|_m^nv)|_m^n(v|_m^nv)))|_m^n((v|_m^nv)|_m^n(v|_m^nv))))} \hfill \cr {} \hfill & = \hfill & {((u|_m^n(v|_m^nv))|_m^n(v|_m^nv),(((v|_m^nv)|_m^n((u|_m^nu)|_m^n(u|_m^nu)))|_m^n((u|_m^nu)|_m^n} \hfill \cr {} \hfill & {} \hfill & {(u|_m^nu)))|_m^n(((v|_m^nv)|_m^n((u|_m^nu)|_m^n(u|_m^nu)))|_m^n((u|_m^nu)|_m^n(u|_m^nu))))} \hfill \cr {} \hfill & = \hfill & {((u|_m^n(v|_m^nv))|_m^n(v|_m^nv),(u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))))} \hfill \cr } is a solution to the set-theoretical Yang-Baxter equation.

Assume that w | m n ( ( u | m n u ) | m n ( v | m n v ) ) = ( ( u | m n w ) | m n ( v | m n w ) ) | m n ( ( u | m n w ) | m n ( v | m n w ) ) w|_m^n((u|_m^nu)|_m^n(v|_m^nv)) = ((u|_m^nw)|_m^n(v|_m^nw))|_m^n((u|_m^nw)|_m^n(v|_m^nw)) holds for all u,v,w ∈ [m,n]. Thus we obtain that S ( u , v ) = ( ( u | m n ( v | m n v ) ) | m n ( v | m n v ) , ( u | m n ( u | m n ( v | m n v ) ) ) | m n ( u | m n ( u | m n ( v | m n v ) ) ) ) = ( ( v | m n v ) | m n ( ( ( u | m n u ) | m n ( u | m n u ) ) | m n ( v | m n v ) ) , ( u | m n ( ( ( u | m n u ) | m n ( u | m n u ) ) | m n ( v | m n v ) ) ) | m n ( u | m n ( ( ( u | m n u ) | m n ( u | m n u ) ) | m n ( v | m n v ) ) ) )                                         ( ( S 2 )   and ( S 1 ) ) = ( ( ( ( u | m n u ) | m n ( v | m n v ) ) | m n ( v | m n ( v | m n v ) ) ) | m n ( ( ( u | m n u ) | m n ( v | m n v ) ) | m n ( v | m n ( v | m n v ) ) ) , ( ( ( u | m n ( u | m n u ) ) | m n ( u | m n v ) ) | m n ( ( u | m n ( u | m n u ) ) | m n ( u | m n v ) ) ) | m n ( ( ( u | m n ( u | m n u ) ) | m n ( u | m n v ) ) | m n ( ( u | m n ( u | m n u ) ) | m n ( u | m n v ) ) ) )                         ( hyp .   and   ( S 1 ) ) = ( ( u | m n u ) | m n ( v | m n v ) , ( u | m n v ) | m n ( u | m n v ) )                                                                                                 ( Lemma   5   ( 1 ) ,   ( 3 ) ,   ( S 1 )   and   ( S 2 ) ) \matrix{ {S(u,v)} \hfill & = \hfill & {((u|_m^n(v|_m^nv))|_m^n(v|_m^nv),(u|_m^n(u|_m^n(v|_m^nv)))|_m^n(u|_m^n(u|_m^n(v|_m^nv))))} \hfill \cr {} \hfill & = \hfill & {((v|_m^nv)|_m^n(((u|_m^nu)|_m^n(u|_m^nu))|_m^n(v|_m^nv)),(u|_m^n(((u|_m^nu)|_m^n} \hfill \cr {} \hfill & {} \hfill & {(u|_m^nu))|_m^n(v|_m^nv)))|_m^n(u|_m^n(((u|_m^nu)|_m^n(u|_m^nu))|_m^n(v|_m^nv))))\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;((S2)\;and(S1))} \hfill \cr {} \hfill & = \hfill & {((((u|_m^nu)|_m^n(v|_m^nv))|_m^n(v|_m^n(v|_m^nv)))|_m^n(((u|_m^nu)|_m^n(v|_m^nv))|_m^n} \hfill \cr {} \hfill & {} \hfill & {(v|_m^n(v|_m^nv))),(((u|_m^n(u|_m^nu))|_m^n(u|_m^nv))|_m^n((u|_m^n(u|_m^nu))|_m^n} \hfill \cr {} \hfill & {} \hfill & {(u|_m^nv)))|_m^n(((u|_m^n(u|_m^nu))|_m^n(u|_m^nv))|_m^n((u|_m^n(u|_m^nu))|_m^n(u|_m^nv))))\;\;\;\;\;\;\;\;\;\;(hyp.\;and\;(S1))} \hfill \cr {} \hfill & = \hfill & {((u|_m^nu)|_m^n(v|_m^nv),(u|_m^nv)|_m^n(u|_m^nv))\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(Lemma\;5\;(1),\;(3), (S1)\;and\;(S2))} \hfill \cr } is a solution to the set-theoretical Yang-Baxter equation.

Theorem 16

Let ( [ m , n ] ; | m n ) ([m,n];|_m^n) be an interval Sheffer stroke basic algebra. If ( u | m n ( v | m n v ) ) | m n ( ( u | m n ( w | m n w ) ) | m n ( u | m n ( w | m n w ) ) ) = u | m n ( ( v | m n ( w | m n w ) ) | m n ( v | m n ( w | m n w ) ) ) (u|_m^n(v|_m^nv))|_m^n((u|_m^n(w|_m^nw))|_m^n(u|_m^n(w|_m^nw))) = u|_m^n((v|_m^n(w|_m^nw))|_m^n(v|_m^n(w|_m^nw))) holds for all u,v,w ∈ [m,n], then S ( u , v ) = ( u | m n ( v | m n v ) , u ) S(u,v) = (u|_m^n(v|_m^nv),u) is a solution to the set-theoretical Yang-Baxter equation.

Proof

S12 and S23 are defined in the following forms: S 12 ( u , v , w ) = ( u | m n ( v | m n v ) , u , w ) ,           S 23 ( u , v , w ) = ( u , v | m n ( w | m n w ) , v ) . {S^{12}}(u,v,w) = (u|_m^n(v|_m^nv),u,w),\;\;\;\;\;{S^{23}}(u,v,w) = (u,v|_m^n(w|_m^nw),v). For all u,v,w ∈ [m,n], we have ( S 12 ∘ S 23 ∘ S 12 ) ( u , v , w ) = ( S 12 ∘ S 23 ) ( S 12 ( u , v , w ) ) = ( S 12 ∘ S 23 ) ( u | m n ( v | m n v ) , u , w ) = S 12 ( S 23 ( u | m n ( v | m n v ) , u , w ) ) = S 12 ( u | m n ( v | m n v ) , u | m n ( w | m n w ) , u ) = ( ( u | m n ( v | m n v ) ) | m n ( ( u | m n ( w | m n w ) ) | m n ( u | m n ( w | m n w ) ) ) , u | m n ( v | m n v ) , u ) = ( u | m n ( ( v | m n ( w | m n w ) ) | m n ( v | m n ( w | m n w ) ) ) , u | m n ( v | m n v ) , u )                                                         ( hyp . ) \matrix{ {({S^{12}} \circ {S^{23}} \circ {S^{12}})(u,v,w)} \hfill & = \hfill & {({S^{12}} \circ {S^{23}})({S^{12}}(u,v,w))} \hfill \cr {} \hfill & = \hfill & {({S^{12}} \circ {S^{23}})(u|_m^n(v|_m^nv),u,w)} \hfill \cr {} \hfill & = \hfill & {{S^{12}}({S^{23}}(u|_m^n(v|_m^nv),u,w))} \hfill \cr {} \hfill & = \hfill & {{S^{12}}(u|_m^n(v|_m^nv),u|_m^n(w|_m^nw),u)} \hfill \cr {} \hfill & = \hfill & {((u|_m^n(v|_m^nv))|_m^n((u|_m^n(w|_m^nw))|_m^n(u|_m^n(w|_m^nw))),u|_m^n(v|_m^nv),u)} \hfill \cr {} \hfill & = \hfill & {(u|_m^n((v|_m^n(w|_m^nw))|_m^n(v|_m^n(w|_m^nw))),u|_m^n(v|_m^nv),u)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(hyp.)} \hfill \cr} and ( S 23 ∘ S 12 ∘ S 23 ) ( u , v , w ) = ( S 23 ∘ S 12 ) ( S 23 ( u , v , w ) ) = ( S 23 ∘ S 12 ) ( ( u , v | m n ( w | m n w ) , v ) = S 23 ( S 12 ( u , v | m n ( w | m n w ) , v ) ) = S 23 ( u | m n ( ( v | m n ( w | m n w ) ) | m n ( v | m n ( w | m n w ) ) ) , u , v ) = ( u | m n ( ( v | m n ( w | m n w ) ) | m n ( v | m n ( w | m n w ) ) ) , u | m n ( v | m n v ) , u ) \matrix{ {({S^{23}} \circ {S^{12}} \circ {S^{23}})(u,v,w)} \hfill & = \hfill & {({S^{23}} \circ {S^{12}})({S^{23}}(u,v,w))} \hfill \cr {} \hfill & = \hfill & {({S^{23}} \circ {S^{12}})((u,v|_m^n(w|_m^nw),v)} \hfill \cr {} \hfill & = \hfill & {{S^{23}}({S^{12}}(u,v|_m^n(w|_m^nw),v))} \hfill \cr {} \hfill & = \hfill & {{S^{23}}(u|_m^n((v|_m^n(w|_m^nw))|_m^n(v|_m^n(w|_m^nw))),u,v)} \hfill \cr {} \hfill & = \hfill & {(u|_m^n((v|_m^n(w|_m^nw))|_m^n(v|_m^n(w|_m^nw))),u|_m^n(v|_m^nv),u)} \hfill \cr} Then S ( u , v ) = ( u | m n ( v | m n v ) , u ) S(u,v) = (u|_m^n(v|_m^nv),u) is a solution to the set-theoretical Yang-Baxter equation in the interval Sheffer stroke basic algebra.

Example 17

Consider an interval Sheffer stroke basic algebra ( M ; | m n ) (M;|_m^n) where the set M = {m, p,r,n} with the Cayley table as below:

| m n |_m^n m p r n

m n n n n
p n r n r
r n n p p
n n r p m
Since ( u | m n ( v | m n v ) ) | m n ( ( u | m n ( w | m n w ) ) | m n ( u | m n ( w | m n w ) ) ) = u | m n ( ( v | m n ( w | m n w ) ) | m n ( v | m n ( w | m n w ) ) ) (u|_m^n(v|_m^nv))|_m^n((u|_m^n(w|_m^nw))|_m^n(u|_m^n(w|_m^nw))) = u|_m^n((v|_m^n(w|_m^nw))|_m^n(v|_m^n(w|_m^nw))) holds for all u,v,w ∈ M, S ( u , v ) = ( u | m n ( v | m n v ) , u ) S(u,v) = (u|_m^n(v|_m^nv),u) is a solution to the set-theoretical Yang-Baxter equation in this algebraic structure.

j.amns.2020.2.00065.tab.001

| m n |_m^n m p r n

m n n n n
p n r n r
r n n p p
n n r p m

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