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Good congruences on weakly U-abundant semigroups

Data publikacji: 20 May 2022
Tom & Zeszyt: AHEAD OF PRINT
Zakres stron: -
Otrzymano: 17 Nov 2021
Przyjęty: 10 Apr 2022
Informacje o czasopiśmie
License
Format
Czasopismo
eISSN
2444-8656
Pierwsze wydanie
01 Jan 2016
Częstotliwość wydawania
2 razy w roku
Języki
Angielski
Abstract

In this paper, we study the congruences on w-U-a semigroups with s-transversals. The g-congruences on a w-U-a semigroup with an s-transversal TV are described abstractly by the e-triple, which consists of equivalences on the structure component parts I, T and Λ. Also, it is shown that the set of all g-congruences on this kind of semigroups forms a complete lattice.

Keywords

MSC 2010

Introduction

In 1982, Blyth and McFadden [2] first proposed the concept of inverse transversals. In 1993, Ei-Qallali [3] introduced ample transversals for abundant semigroups. Recently, Wang introduced E-transversals for E-semiabundant semigroups. According to Howie, the test of whether a structure theory for semigroups is truly satisfactory is to see if it could enable one to give an explicit description of the congruences, or not [6]. After the concept of inverse transversals was given, more and more authors initiated the study of congruencies for regular semigroups with inverse transversals and obtained some interesting results [1, 4, 10,11,12]. Recently, several researchers began to show an interest in congruences for abundant semigroups with ample transversals and obtained some meaningful results [8, 9, 15]. w-U-a semigroups (E-semiabundant semigroups) are a generalisation of an abundant semigroup, and include the class of abundant semigroups as a proper sub-class. The application of congruence theory on w-U-a semigroups is far more complex than that on regular (abundant) semigroups. For example, a homomorphic image of a w-U-a semigroup is not necessarily a w-U-a semigroup but a similar statement is true for a regular semigroup. This property happens to be very important in the study of semigroups. So we consider the g-congruences on a w-U-a semigroup with an s-transversal since this kind of congruence has the above-mentioned property [13].

In this paper, we provide a construction method for g-congruences on w-U-a semigroups with (C) and E-transversals, and prove that the set of all g-congruences on these kinds of semigroups is a complete lattice.

Preliminaries

Let E(S) be the set of all idempotents of a semigroup S, and U be any subset of E(S).

For any a,bS, define ˜U {\tilde {\cal L}_U} and ˜U {\tilde {\cal R}_U} on S are as follows: a˜Ubifandonlyif(eU)(ae=abe=b);a˜Ubifandonlyif(eU)(ea=aeb=b). \matrix{ {a{{\tilde {\cal L}}_U}b\;{\rm{if}}{\kern 1pt} {\rm{and}}{\kern 1pt} {\rm{only}}{\kern 1pt} {\rm{if}}{\kern 1pt} \left( {\forall e \in U} \right)\left( {ae = a \Leftrightarrow be = b} \right);} \hfill \cr {a{{\tilde {\cal R}}_U}b\;{\rm{if}}{\kern 1pt} {\rm{and}}{\kern 1pt} {\rm{only}}{\kern 1pt} {\rm{if}}{\kern 1pt} \left( {\forall e \in U} \right)\left( {ea = a \Leftrightarrow eb = b} \right).} \hfill \cr } For U = E(S), we use ˜U {\tilde {\cal L}_U} and ˜U {\tilde {\cal R}_U} for ˜ \tilde {\cal L} and ˜ \tilde {\cal R} , respectively. It is easy to verify that *˜U {\cal L} \subseteq {{\cal L}^*} \subseteq {\tilde {\cal L}_U} and *˜U {\cal R} \subseteq {{\cal R}^*} \subseteq {\tilde {\cal R}_U} . For U = E(S) and the regular elements a and b, we have =*=˜U {\cal L} = {{\cal L}^*} = {\tilde {\cal L}_U} and =*=˜U {\cal R} = {{\cal R}^*} = {\tilde {\cal R}_U} . S is called weakly U-abundant (w-U-a for short) if each ˜U {\tilde {\cal R}_U} -class and each ˜U {\tilde {\cal L}_U} -class of S contain an idempotent of U. We always write SU instead of S. We call a w-U-a semigroup SU satisfying the congruence condition (c-c) if ˜U {\tilde {\cal L}_U} is a right congruence and ˜U {\tilde {\cal R}_U} is a left congruence ( ˜U {\tilde {\cal L}_U} and ˜U {\tilde {\cal R}_U} are not always right and left congruences, respectively); consequently, we call SU a w-U-a with (C). Further, if U forms a semilattice, then we call SU an Ehresmann semigroup (E-semigroup for short). For an E-semigroup SU and for any aS, there exists a unique idempotent eU such that e˜Ua e{\tilde {\cal L}_U}a , and a unique idempotent fU such that f˜Ua f{\tilde {\cal R}_U}a . We denote e and f by a* and a+, respectively.

For a semigroup denoted by another letter, say Q, we denote the set of its idempotents by E(Q), the relations ˜ \tilde {\cal R} and ˜ \tilde {\cal L} on QU by ˜U(Q) {\tilde {\cal R}_U}(Q) and ˜U(Q) {\tilde {\cal L}_U}(Q) , and the ˜ \tilde {\cal R} -class and the ˜ \tilde {\cal L} -class of an element a by ˜Ua(Q) \tilde {\cal R}_U^a(Q) and ˜Ua(Q) \tilde {\cal L}_U^a(Q) , respectively. The following results are immediately apparent from the definition of ˜U {\tilde {\cal L}_U} . Of course, a dual result holds for ˜U {\tilde {\cal R}_U} .

Lemma 1

[5] Let aS and e, fUE(S). Then

1. e˜Ua(fU)ae=a e{\tilde {\cal L}_U}a \Leftrightarrow (\forall f \in U)ae = a and a f = a implies e f = e;

2. efe˜Uf e{\cal L}f \Leftrightarrow e{\tilde {\cal L}_U}f and efe˜Uf e{\cal R}f \Leftrightarrow e{\tilde {\cal R}_U}f .

Lemma 2

[5] Let SU be an E-semigroup. Then

1. (∀a,bS)(ab)* = (a* b)* and (ab)+ = (ab+)+;

2. (a,bS)a˜Uba+=b+ (\forall a,b \in S)a{\tilde {\cal R}_U}b \Leftrightarrow {a^ + } = {b^ + } , a˜Uba*=b* a{\tilde {\cal L}_U}b \Leftrightarrow {a^*} = {b^*} .

Let TV be a w-V-a semigroup and SU be a w-U-a semigroup. We can call TV a w-V-a subsemigroup of SU if TS and VUT. For any aT, if there exist e, fV such that e˜Ua˜Uf e{\tilde {\cal L}_U}a{\tilde {\cal R}_U}f , then we call TV a ∼ -w-V-a subsemigroup of SU.

Lemma 3

[14] If TV is a ∼ -w-V-a subsemigroup of a w-U-a semigroup SU, then ˜V(T×T)=˜U(T×T) {\tilde {\cal L}_V} \cap (T \times T) = {\tilde {\cal L}_U} \cup (T \times T) and ˜V(T×T)=˜U(T×T), {\tilde {\cal R}_V} \cap (T \times T) = {\tilde {\cal R}_U} \cup (T \times T), and hence, if SU satisfies c-c, then, so dose TV.

Call a-w-V-a subsemigroup TV of a w-U-a semigroup SU with (C) a-E-subsemigroup if V forms a semilattice. Call a-E-subsemigroup TV of SU with (C) an Ehresmann transversal (E-transversal for short) of SUif(xS,e,fU,!x¯T)x=ex¯f {S_U}\;{\rm{if}}\;\left( {\forall x \in S,\exists e,f \in U,\exists !\overline x \in T} \right)x = e\overline x f , where x+ℒ e and x+e {x^ + }{\cal L}e . Based on the work of Yang [14], x can be used to determine the e and f uniquely. So, we denote e by ex and f by fx and put I = {exS} and Λ = {fxS}.

Lemma 4

[14] Let SU be a w-U-a semigroup with (C) and an E-transversal TV. Then

1. if xT, then x¯*f {\overline x ^*}{\cal R}f , ex = x+ and fx = x*;

2. if xV, then x¯=x \overline x = x ;

3. I ∩ Λ = V ;

4. I = {eU : (∃ℓ ∈ V)eℒ ℓ} and Λ = {fU : (∃hV) f ℒ h};

5. (∀eI,f ∈ Λ)ee = e and ff = f;

6. (∀eI, ∃!ℓ ∈ V)eℒ; (∀ f ∈ Λ, ∃!hV) f ℛ h.

A subset T of S is called a q-ideal (b-ideal) of S if STT ST (T STT). For a w-U-a semigroup SU, we have the following result.

Lemma 5

Let TV be an E-transversal of a w-U-a semigroup SU. Then TV is a q-ideal of SU if and only if TV is a b-ideal of SU.

Proof

Suppose that TV is a q-ideal, xT ST, then there exist a, bT, bS such that x = abc = ab · c = a · bcT. Thus TV is a b-ideal of SU. Conversely, if TV is a b-ideal of SU, then, for any xSTT S, there exist a, bS, c, dT such that x¯=x=ex=fx \overline x = x = {e_x} = {f_x} Thus TV is a q-ideal of SU.

We call an E-transversal TV of a w-U-a semigroup SU a q-ideal E-transversal if TV is a q-ideal of SU. We call an q-ideal E-transversal TV of a w-U-a semigroup SU strong (s-transversal for short) if V is an order ideal of E(T), and the idempotents of both IT and T Λ are all sets of U.

Define φ and ψ as follows: x=ac=db=eaa¯fac=debb¯fb=d+eaa¯facT. x = ac = db = {e_a}\overline a {f_a}c = d{e_b}\overline b {f_b} = {d^ + } \cdot {e_a}\overline a {f_a} \cdot c \in T. By Lemma 4, (6), both φ and ψ are well defined, and φ |v = ψ|v = 1V.

Lemma 6

[14] Let TV be a s-transversal of a w-U-a semigroup SU with (C). Then:

1. (∀ fV,eI) f eRegU (S) ⇒ f e = f · eφ;

2. (∀eV,f ∈ Λ) f eRegU (S) ⇒ f e = f ψ · e.

Let V be a semilattice and A and B be non-empty sets. We suppose that V, A and B are mutually disjoint. Putting I = VA and Λ = VB, it is easy to see that I ∩ Λ = V.

A (V,I,Λ)-system means that V, I, λ thus satisfying the conditions (B1)–(B3):

(B1) There exist two mappings φ : IV and ψ : Λ → V such that φ:IV,eeφ=,wheree;ψ:ΛV,ffψ=h,wherefh. \matrix{ {\varphi :\;I \to V,\;e \mapsto e\varphi = \ell ,\;{\rm{where}}\;e{\cal L}\ell ;} \hfill \cr {\psi :\;\Lambda \to V,\;f \mapsto f\psi = h,\;{\rm{where}}\;f{\cal R}h.} \hfill \cr }

(B2) Mappings ⊗: I × VI and: V × Λ → Λ satisfying φ|v=ψ|v=1|V. \varphi \left| {v = \psi } \right|v{ = 1|_V}.

(B3) For any xI, y ∈ Λ, sV, we have (s,tV,xI)x(st)=(xs)t,(s,tV,yΛ)(st)*y=s*(t*y). \matrix{ {(\forall s,t \in V,\forall x \in I)x \otimes (st) = (x \otimes s) \otimes t,} \hfill \cr {(\forall s,t \in V,\forall y \in \Lambda )(st)*y = s*(t*y).} \hfill \cr } that is, φ, ψ, ⊗,* are compatible.

In the rest of this paper, we use E to denote the set of all idempotents of W. For a semigroup S, we denote the set of all regular elements of S by Reg(S), and define xxφ=x,yψ*y=y, {x \otimes x\varphi = x,y\psi *y = y,}

Lemma 7

[14] Suppose that there is a (V, I, Λ)-system, TV is an E-semigroup in which V is an order ideal of E(T), and P is a Λ × I matrix over T satisfying (denote Pf,e by [f, e])

(B4) (∀a,bV)a[f, e]b = [a * f,eb],

(B5) (∀ f ∈ Λ,∀eI)[f ψ,eφ] = f ψ · eφ; f, f ψ] = f ψ; [eφ,e] = eφ and (xs)φ=xφs=xφs,(s*y)ψ=s*yψ=syψ, {(x \otimes s)\varphi = x\varphi \otimes s = x\varphi \cdot s,{\kern 1pt} (s*y)\psi = s*y\psi = s \cdot y\psi ,}

Put VU(a)={a'V(a)|aa',a'aU}andRegU(S)={aReg(S)|VU(a)ϕ}. {V_U}(a) = \{ {a^\prime} \in V(a)|a{a^\prime},{a^\prime}a \in U\} {\kern 1pt} {\rm{and}}{\kern 1pt} {{\rm{Reg}}_U}(S) = \{ a \in {\rm{Reg}}(S)|{V_U}(a) \ne \phi \} . with a multiplication [fψ,e]RegV(T)[fψ,e]=fψeφ,[f,eφ]RegV(T)[f,eφ]=fψeφ. \matrix{ {[f\psi ,e] \in {{{Reg}}_V}(T) \Rightarrow [f\psi ,e] = f\psi \cdot e\varphi ,} \hfill \cr {[f,e\varphi ] \in {{{Reg}}_V}(T) \Rightarrow [f,e\varphi ] = f\psi \cdot e\varphi .} \hfill \cr } where a = x[f,g]yT. Then WE is a w-E-a semigroup with (C) and an s-transversal isomorphic to TV. Conversely, every w-U-a semigroup SU with (C) and an s-transversal TV can be constructed in the above manner.

For other notations and terminologies given in this paper, the reader is referred to other papers in the literature [6, 13, 14].

Good congruences

In this section, we describe the g-congruences on a w-U-a semigroup SU with (C) and an s-transversal TV. Let V, A and B be mutually disjoint sets, I = VA and Λ = VB (hence I ∩ Λ = V). Assume that V, I, Λ form an (V, I, Λ)-system. We first give the following definitions.

Definition 1

A congruence ρ on SU is called a good congruence (g-congruence for short) if, for any x, yT, (x,y) ∈ ρ implies (x+,y+) ∈ ρ and (x*, y*) ∈ ρ; an equivalence ρ on I[Λ] is called a normal equivalence (n-equivalence for short), if for all m, nV, iI[j ∈ Λ], mρn implies (im)ρ(in)[(m * j)ρ(n * j)]. Let ρI and ρΛ be n-equivalences on I and Λ, respectively, and ρT is a g-congruence on T. We call (ρI, ρT, ρΛ) an equivalence triple (e-triple for short) on I × T × Λ if the following conditions hold:

(C.1) ρI|V = ρΛ|V = ρT |V ;

(C.2) (∀ j ∈ Λ)(∀e,gI)Ig ⇒ [j,e]ρT [j,g];

(C.3) (∀iI)(∀ f, h ∈ Λ) f ρΛh ⇒ [f, i]ρT [h,i].

Define a relation ρ(ρITΛ) on the semigroup W in Lemma 7 by W={(e,x,f)I×T×Λ:eφ=x+,fψ=x*} W = \{ (e,x,f) \in I \times T \times \Lambda :e\varphi = {x^ + },f\psi = {x^*}\}

Theorem 1

Let W be a w-U-a semigroup SU with (C) and a s-transversal TV as in Lemma 7, and (ρI, ρT, ρΛ) be an e-triple on T. Then ρ(ρITΛ) is a g-congruence on W.

Conversely, every g-congruence on W can be constructed in the above-mentioned manner.

Proof

For any e-triple (ρITΛ) ∈ W. It is clear that ρ(ρITΛ) is an equivalence. Supposing that (i,z, j),(e,x, f),(g,y,h) ∈ W, and (e,x, f)ρ(ρITΛ) (g, y, h), then, by (*), T y, Ig and f ρΛh, and so (e,x,f)(g,y,h)=(ea+,a,a**h) (e,x,f)(g,y,h) = (e \otimes {a^ + },a,{a^*}*h) where m = z[j,e]x and n = z[j,g]y. Moreover, we have [j,e]ρT [j,g], since Ig, (C.2) and T y, and so T n. On the other hand, since ρT is a g-congruence, we have m+ρT n+ and so, by (C.1), we get m+ρIn+. Therefore, (im+)ρI(in+). Similarly, we can verify (m* * f)ρΛ(n* * h). Thus, we have the following relation (e,x,f)ρ(ρI,ρT,ρΛ)(g,y,h)eρIg,xρTy,fρΛh. (e,x,f){\rho ^{({\rho ^I},{\rho ^T},{\rho ^\Lambda })}}(g,y,h) \Leftrightarrow e{\rho ^I}g,x{\rho ^T}y,f{\rho ^\Lambda }h. that is to say, (i,z,j)(e,x,f)=(im+,m,m**f),(i,z,j)(g,y,h)=(in+,n,n**h), \matrix{ {(i,z,j)(e,x,f) = (i \otimes {m^ + },m,{m^*}*f),} \hfill \cr {(i,z,j)(g,y,h) = (i \otimes {n^ + },n,{n^*}*h),} \hfill \cr } Further, we can prove (e,x, f)(i,z, j)ρ(ρITΛ) (g,y,h)(i,z, j) dually. So, to summarise, we proved that ρ(ρITΛ) is a congruence.

Supposing that (x+,x,x*),(y+,y,y*) ∈ W′ and (x+,x,x*)ρ(ρITΛ) (y+,y,y*), then by (*) (im+,m,m**f)ρ(ρI,ρT,ρΛ)(in+,n,n**h), (i \otimes {m^ + },m,{m^*}*f){\rho ^{({\rho ^I},{\rho ^T},{\rho ^\Lambda })}}(i \otimes {n^ + },n,{n^*}*h), Moreover, we have x+ρT y+ since ρT is a g-congruence, and so by (C.1), x+ρΛy+. Thus, we have the following relation (i,z,j)(e,x,f)ρ(ρI,ρT,ρΛ)(i,z,j)(g,y,h). (i,z,j)(e,x,f){\rho ^{({\rho ^I},{\rho ^T},{\rho ^\Lambda })}}(i,z,j)(g,y,h). Similarly, we have the relation x+ρIy+,xρTy,x*ρΛx*. {x^ + }{\rho ^I}{y^ + },x{\rho ^T}y,{x^*}{\rho ^\Lambda }{x^*}. To sum up, it follows that ρ(ρITΛ) is a g-congruence.

Conversely, let ρ be a g-congruence on W. Define the equivalences on I, Λ and T, respectively:

(E1) (∀e,gI,eφ = x+, = y+)Ig ⇔ (e,x+,x+)ρ(g,y+,y+);

(E2) (∀ f, h ∈ Λ, f ψ = x*,hψ = y*) f ρΛh ⇔ (x*, x*, f)ρ(y*, y*, h);

(E3) (∀x,yT,xρT y ⇔ (x+, x, x*)ρ(y+,y, y*).

We have demonstrated that ρI and ρΛ are equivalences on I and Λ, respectively, and that ρT is a g-congruence on T since ρ is a g-congruence on W.

For any i,kV, if Ik, then we have (i,i,i),(k,k,k) ∈ T and (i,i,i)ρ(k,k,k). Moreover, we have (e,x+,x+) (i,i,i)ρ(e,x+,x+)(k,k,k) since it is a g-congruence, and so (x+,x+,x+)ρ(ρI,ρT,ρΛ)(y+,y+,y+). ({x^ + },{x^ + },{x^ + }){\rho ^{({\rho ^I},{\rho ^T},{\rho ^\Lambda })}}({y^ + },{y^ + },{y^ + }). That is to say, (x*,x*,x*)ρ(ρI,ρT,ρΛ)(y*,y*,y*). ({x^*},{x^*},{x^*}){\rho ^{({\rho ^I},{\rho ^T},{\rho ^\Lambda })}}({y^*},{y^*},{y^*}). Thus we get (ei)ρI(ek) and so ρI is an n-equivalence. By a similar argument, ρΛ it is also an n-equivalence.

Further we have the cases 1–3:

Case 1. Obviously, ρI|V = ρΛ|V = ρT |V. So (C.1) holds.

Case 2. For any e,gI, if = x+, = y+ and Ig, then, by (E1), we have (e,x+,x+)ρ(g,y+,y+). So, for any (z*, z*, j) ∈ W, we have (e(x+[x+,i]i)+,x+[x+,i]i,(x+[x+,i]i)**i)ρ(e(x+[x+,k]k)+,x+[x+,k]k,(x+[x+,k]k)**k). \matrix{ {\left( {e \otimes {{\left( {{x^ + }\left[ {{x^ + },i} \right]i} \right)}^ + },{x^ + }\left[ {{x^ + },i} \right]i,{{\left( {{x^ + }\left[ {{x^ + },i} \right]i} \right)}^*}*i} \right)\rho } \hfill \cr {\left( {e \otimes {{\left( {{x^ + }\left[ {{x^ + },k} \right]k} \right)}^ + },{x^ + }\left[ {{x^ + },k} \right]k,{{\left( {{x^ + }\left[ {{x^ + },k} \right]k} \right)}^*}*k} \right).} \hfill \cr }

That is to say, (ei,x+i,x+i)ρ(ek,x+k,x+k). \left( {e \otimes i,{x^ + }i,{x^ + }i} \right)\rho \left( {e \otimes k,{x^ + }k,{x^ + }k} \right). Thus, we can get z* [j,e]x+ρS z* [j, g]y+. Moreover, by (B3) and (B4), we have z*[j,e]x+ = [j,e] and z*[j,g]y+ = [j,g]. Thus, we have (z*,z*,j)(e,x+,x+)ρ(z*,z*,j)(g,y+,y+). ({z^*},{z^*},j)(e,{x^ + },{x^ + })\rho ({z^*},{z^*},j)(g,{y^ + },{y^ + }). So (C.2) holds.

Case 3. Similar to case 2, (C.3) also holds.

In conclusion, we have that (ρI, ρT, ρΛ) is an e-triple on W.

By the direct part, ρ(ρITΛ) is a g-congruence. On the other hand, for any (e,x, f),(g,y,h) ∈ W, (e,x, f) ρ(ρITΛ) (g,y,h), we have (z*(z*[j,e]x+)+,z*[j,e]x+,(z*[j,e]x+)**x+)ρ(z*(z*[j,g]y+)+,z*[j,g]y+,(z[j,g]y+)**y+). \matrix{ {\left( {{z^*} \otimes {{\left( {{z^*}\left[ {j,e} \right]{x^ + }} \right)}^ + },{z^*}\left[ {j,e} \right]{x^ + },{{\left( {{z^*}\left[ {j,e} \right]{x^ + }} \right)}^*}*{x^ + }} \right)\rho } \hfill \cr {\left( {{z^*} \otimes {{\left( {{z^*}\left[ {j,g} \right]{y^ + }} \right)}^ + },{z^*}\left[ {j,g} \right]{y^ + },{{\left( {z\left[ {j,g} \right]{y^ + }} \right)}^*}*{y^ + }} \right).} \hfill \cr } It follows that [j,e]ρT[j,g]. [j,e]{\rho _T}[j,g]. Further eρIg,xρTy,fρΛh. e{\rho _I}g,{\kern 1pt} x{\rho _T}y,f{\rho _\Lambda }h. That is to say, (e,x, f)ρ (g,y,h) and so ρ(ρITΛ)ρ. Obviously, ρρ(ρITΛ), and thus we have ρ(ρITΛ) = ρ.

Next, we give an example of a g-congruence on a w-U-a semigroup QU with (C) and an s-transversal.

Example 1

Let Q = {e,g,h,w, f} with the following multiplication table

e g h w f
e e g e g g
g g g g g g
h h g h g g
w w g w g g
f g g w w f

Associativity may be checked directly. And it is easy to verify that E(Q) = {e,g,h, f} and that T′ = {w,e, f,g} is a subsemigroup of Q. Taking U′= V′ = {e,g, f}, we then have

(e,x+,x+)ρ(g,y+,y+),(x+,x,x*)ρ(y+,y,y*),(x*,x*,f)ρ(y*,y*,h). \left( {e,{x^ + },{x^ + }} \right)\rho \left( {g,{y^ + },{y^ + }} \right),\left( {{x^ + },x,{x^*}} \right)\rho \left( {{y^ + },y,{y^*}} \right),\left( {{x^*},{x^*},f} \right)\rho \left( {{y^*},{y^*},h} \right). ;

(e,x+,x+)(x+,x,x*),(x*,x*,f)ρ(g,y+,y+),(y+,y,y*)(y*,y*,h). \left( {e,{x^ + },{x^ + }} \right)\left( {{x^ + },x,{x^*}} \right),\left( {{x^*},{x^*},f} \right)\rho \left( {g,{y^ + },{y^ + }} \right),\left( {{y^ + },y,{y^*}} \right)\left( {{y^*},{y^*},h} \right). ;

e˜Uh˜Uw˜Uf e{\tilde {\cal L}_U}h{\tilde {\cal L}_U}w{\tilde {\cal R}_U}f ; L˜V'g(T')=L˜U'g(Q)=R˜V'g(T')=R˜U'g(Q)={g} \tilde L_{{V^\prime}}^g({T^\prime}) = \tilde L_{{U^\prime}}^g(Q) = \tilde R_{{V^\prime}}^g({T^\prime}) = \tilde R_{{U^\prime}}^g(Q) = \{ g\} ;

L˜V'w(T')={w,e} \tilde L_{{V^\prime}}^w({T^\prime}) = \{ w,e\} .

It is easy to check that QU′ is a w-U′-a semigroup QU′ with (C) and an s-transversal L˜U'w(Q)={w,e,h} \tilde L_{{U^\prime}}^w(Q) = \{ w,e,h\} .

It is easy to verify that the identical relation is a g-congruence.

Let gC(W) denote the set of all g-congruences on W and eT (W) denote the set of all e-triples on W constructed as in Theorem 1. Then we have the following results.

Lemma 8

If R˜V'w(T')=R˜U'w(Q)={w,f} \tilde R_{{V^\prime}}^w({T^\prime}) = \tilde R_{{U^\prime}}^w(Q) = \{ w,f\} , (ρ1I,ρ1T,ρ1),(ρ2I,ρ2T,ρ2Λ)eT(W) (\rho _1^I,\rho _1^T,{\rho _1}),(\rho _2^I,\rho _2^T,\rho _2^\Lambda ) \in eT(W) , then ρ(ρ1I,ρ1T,ρ1Λ)ρ(ρ2I,ρ2T,ρ2Λ)ρ1Iρ2I,ρ1Tρ2T,ρ1Λρ2Λ. {\rho ^{(\rho _1^I,\rho _1^T,\rho _1^\Lambda )}} \subseteq {\rho ^{(\rho _2^I,\rho _2^T,\rho _2^\Lambda )}} \Leftrightarrow \rho _1^I \subseteq \rho _2^I,\rho _1^T \subseteq \rho _2^T,\rho _1^\Lambda \subseteq \rho _2^\Lambda .

Proof

Sufficiency. Let us suppose that e,gI such that eρ1Ig e\rho _1^Ig , = x+ and = y+. By the proof of Theorem 1, we have ((e,x+,x+),(g,y+,y+))ρ(ρ1I,ρ1T,ρ1Λ)ρ(ρ2I,ρ2T,ρ2Λ). \left( {\left( {e,{x^ + },{x^ + }} \right),\left( {g,{y^ + },{y^ + }} \right)} \right) \in {\rho ^{(\rho _1^I,\rho _1^T,\rho _1^\Lambda )}} \subseteq {\rho ^{(\rho _2^I,\rho _2^T,\rho _2^\Lambda )}}. It follows that eρ2Ig e\rho _2^Ig , and so ρ1Iρ2I \rho _1^I \subseteq \rho _2^I . Similarly, we can get ρ1Λρ2Λ \rho _1^\Lambda \subseteq \rho _2^\Lambda . On the other hand, if xρ1Ty x\rho _1^Ty , we have x+ρ1Ty+ {x^ + }\rho _1^T{y^ + } and x*ρ1Ty* {x^*}\rho _1^T{y^*} since ρ1T \rho _1^T is a g-congruence on T. Moreover, by (C.1), we have ((x+,x,x*),(y+,y,y*))ρ(ρ1I,ρ1T,ρ1Λ)ρ(ρ2I,ρ2T,ρ2Λ). \left( {\left( {{x^ + },x,{x^*}} \right),\left( {{y^ + },y,{y^*}} \right)} \right) \in {\rho ^{(\rho _1^I,\rho _1^T,\rho _1^\Lambda )}} \subseteq {\rho ^{(\rho _2^I,\rho _2^T,\rho _2^\Lambda )}}. Hence, xρ2Ty x\rho _2^Ty .

Necessity. It is obvious.

Define relation ≤ on eT (W) by (ρ1I,ρ1T,ρ1Λ)(ρ2I,ρ2T,ρ2Λ)ρ1Iρ2I,ρ1Tρ2T,ρ1Λρ2Λ. \left( {\rho _1^I,\rho _1^T,\rho _1^\Lambda } \right) \le \left( {\rho _2^I,\rho _2^T,\rho _2^\Lambda } \right) \Leftrightarrow \rho _1^I \subseteq \rho _2^I,\rho _1^T \subseteq \rho _2^T,\rho _1^\Lambda \subseteq \rho _2^\Lambda .

Then eT (W) is a partial ordered set with respect to ≤. By Theorem 1 and Lemma 8, gC(W) and eT (W) are isomorphic as partial ordered sets. For any Ω ⊆ gC(W), VρΩgρ={σgC(W)|ρσ,forallρΩ} \mathop V\limits_{\rho \in \Omega }^g \rho = \cap \{ \sigma \in gC(W)|\rho \subseteq \sigma ,\;{\rm{for}}\;{\rm{all}}\;\rho \in \Omega \} , that is, the smallest g-congruence on W containing every congruence of Ω.

Proposition 1

Let Ω ⊆ gC(W) and Wρ = ρI, ρT, ρΛ, where ρ ∈ Ω. Then WρΩρ=(ρΩρI,ρΩρT,ρΩρΛ) {W_{\mathop \cap \limits_{\rho \in \Omega } \rho }} = \left( {\mathop {\bigcap }\limits_{\rho \in \Omega } {\rho ^I},\mathop {\bigcap }\limits_{\rho \in \Omega } {\rho ^T},\mathop {\bigcap }\limits_{\rho \in \Omega } {\rho ^\Lambda }} \right) and WVρΩgρ=(VρΩgρI,VρΩgρT,VρΩgρΛ). {W_{\mathop V\limits_{\rho \in \Omega }^g \rho }} = \left( {\mathop V\limits_{\rho \in \Omega }^g {\rho ^I},\mathop V\limits_{\rho \in \Omega }^g {\rho ^T},\mathop V\limits_{\rho \in \Omega }^g {\rho ^\Lambda }} \right).

Proof

The first equality is pretty obvious, and so we just need to prove the second equality.

For any e, gI, x, yT, if = x+, = y+ and e(VρΩgρ)Ig e{\left( {\mathop V\nolimits_{\rho \in \Omega }^g \rho } \right)^I}g , then we have i=(e,x+,x+)VρΩgρ(g,y+,y+)=j. i = (e,{x^ + },{x^ + })\mathop V\limits_{\rho \in \Omega }^g \rho (g,{y^ + },{y^ + }) = j. Hence, there exist ρi ∈ Ω and ai = (ei,xi, fi) ∈ W such that 1a1ρ2a2 ···an−1ρnj. This implies that i+ρ1a1+ρ2a2+an1+ρnj+ {i^ + }{\rho _1}a_1^ + {\rho _2}a_2^ + \cdots a_{n - 1}^ + {\rho _n}{j^ + } and eρ1Ie1ρ2Ie2en1ρnIg e\rho _1^I{e_1}\rho _2^I{e_2} \cdots {e_{n - 1}}\rho _n^Ig . Thus we have (VρΩgρ)IVρΩgρI. {\left( {\mathop V\limits_{\rho \in \Omega }^g \rho } \right)^I} \subseteq \mathop V\limits_{\rho \in \Omega }^g {\rho ^I}. The converse is obvious. Similarly, we have (VρΩgρ)Λ=VρΩgρΛ {\left( {\mathop V\nolimits_{\rho \in \Omega }^g \rho } \right)^\Lambda } = \mathop V\nolimits_{\rho \in \Omega }^g {\rho ^\Lambda }

Next, assume that x(VρΩgρ)Ty x{\left( {\mathop V\limits_{\rho \in \Omega }^g \rho } \right)^T}y , then we have s=(x+,x,x*)VρΩgρ(y+,y,y*)=t s = \left( {{x^ + },x,{x^*}} \right)\mathop V\limits_{\rho \in \Omega }^g \rho \left( {{y^ + },y,{y^*}} \right) = t . Further, there exist si=(xi+,xi,xi*)W {s_i} = \left( {x_i^ + ,{x_i},x_i^*} \right) \in W and ρi ∈ Ω such that 1s1ρ2s2 ···ρn−1sn−1ρnt and so xρ1Tx1ρ2Tx2ρn1Txn1ρnTy x\rho _1^T{x_1}\rho _2^T{x_2} \cdots \rho _{n - 1}^T{x_{n - 1}}\rho _n^Ty . Hence, we have (VρΩgρ)TVρΩgρT. {\left( {\mathop V\limits_{\rho \in \Omega }^g \rho } \right)^T} \subseteq \mathop V\limits_{\rho \in \Omega }^g {\rho ^T}. On the other hand, VρΩgρT(VρΩgρ)T \mathop V\nolimits_{\rho \in \Omega }^g {\rho ^T} \subseteq {\left( {\mathop V\nolimits_{\rho \in \Omega }^g \rho } \right)^T} is clear and so (VρΩgρ)TVρΩgρT {\left( {\mathop V\nolimits_{\rho \in \Omega }^g \rho } \right)^T} \subseteq \mathop V\nolimits_{\rho \in \Omega }^g {\rho ^T} .

To sum up, we obtain the following theorem.

Theorem 2

Let W be constructed as in Lemma 7. Then eT (W) forms a complete lattice with respect toand gC(W) is isomorphic to eT (W) as a complete lattice.

j.amns.2021.2.00193.tab.001

e g h w f
e e g e g g
g g g g g g
h h g h g g
w w g w g g
f g g w w f

Almeida Santos, M. H. (1998). Inverse transversal congruence extensions. Comm. Algebra. 26(3):889–898. Almeida SantosM. H. 1998 Inverse transversal congruence extensions Comm. Algebra. 26 3 889 898 10.1080/00927879808826171 Search in Google Scholar

Blyth, T. S., McFadden, R. B. (1982). Regular semigroups with a multiplicative inverse transversal. Proc. Roy. Soc. Edinburgh. 92A:253–270. BlythT. S. McFaddenR. B. 1982 Regular semigroups with a multiplicative inverse transversal Proc. Roy. Soc. Edinburgh. 92A 253 270 10.1017/S0308210500032522 Search in Google Scholar

El-Qallali, A. (1993). Abundant semigroups with a multiplicative type A transversal. Semigroup Forum 47:327–340. El-QallaliA. 1993 Abundant semigroups with a multiplicative type A transversal Semigroup Forum 47 327 340 10.1007/BF02573770 Search in Google Scholar

Feng, Y. Y., Wang, L. M. (2012). The K-relation on the congruence lattice of a regular semigroup with an inverse transversal. Acta Math. Sin. (Engl. Ser.) 28(5):1061–1074. FengY. Y. WangL. M. 2012 The K-relation on the congruence lattice of a regular semigroup with an inverse transversal Acta Math. Sin. (Engl. Ser.) 28 5 1061 1074 10.1007/s10114-011-9112-0 Search in Google Scholar

Gomes, G., Gould, V. (2008). Fundamental semigroups having a band of idempotents. Semigroup Forum 77:279–299. GomesG. GouldV. 2008 Fundamental semigroups having a band of idempotents Semigroup Forum 77 279 299 10.1007/s00233-007-9041-5 Search in Google Scholar

Howie, J. M. (1967). An introduction to semigroup theory. Academic Press, London. HowieJ. M. 1967 An introduction to semigroup theory Academic Press London Search in Google Scholar

Kong, X. J. (2014). On generated orthodox transversals. Comm. Algebra 42:1431–1447. KongX. J. 2014 On generated orthodox transversals Comm. Algebra 42 1431 1447 10.1080/00927872.2012.730592 Search in Google Scholar

Luo, Y. F., Zhao, H., Chen, J. F. (2004). Good congruences on abundant semigroups with multiplicative adequate transversals. Southeast Asian Bull. Math. 28(6):1063–1074. LuoY. F. ZhaoH. ChenJ. F. 2004 Good congruences on abundant semigroups with multiplicative adequate transversals Southeast Asian Bull. Math. 28 6 1063 1074 Search in Google Scholar

Ni, X. F., Luo, Y. F., Chao, H. Z. (2011). Good congruences on abundant semigroups with multiplicative quasi-adequate transversals. Semigroup Forum 82(2):324–337. NiX. F. LuoY. F. ChaoH. Z. 2011 Good congruences on abundant semigroups with multiplicative quasi-adequate transversals Semigroup Forum 82 2 324 337 10.1007/s00233-010-9286-2 Search in Google Scholar

Tang, X. L., Wang, L. M. (1995). Congruences on regular semigroups with inverse transversals. Comm. Algebra. 23:4157–4171. TangX. L. WangL. M. 1995 Congruences on regular semigroups with inverse transversals Comm. Algebra. 23 4157 4171 10.1080/00927879508825455 Search in Google Scholar

Wang, A. F., Wang, L. L. (2016). Congruences on regular semigroups with Q-inverse transversals. Ukrainian Math. J. 68(6):972–980. WangA. F. WangL. L. 2016 Congruences on regular semigroups with Q-inverse transversals Ukrainian Math. J. 68 6 972 980 10.1007/s11253-016-1270-x Search in Google Scholar

Wang, L. M. (1995). On congruence lattices of regular semigroups with Q-inverse transversals. Semigroup Forum 50:141–160. WangL. M. 1995 On congruence lattices of regular semigroups with Q-inverse transversals Semigroup Forum 50 141 160 10.1007/BF02573513 Search in Google Scholar

Wang, S. F. (2018). On E-semiabundant semigroups with a multiplicative restriction transversal. Studia Sci. Math. Hungarica 55:153–173. WangS. F. 2018 On E-semiabundant semigroups with a multiplicative restriction transversal Studia Sci. Math. Hungarica 55 153 173 10.1556/012.2018.55.2.1374 Search in Google Scholar

Yang, D. D. (2018). Weakly U-abundant semigroups with strong Ehresmann transversals. Math. Slovaca 68(3):549–562. YangD. D. 2018 Weakly U-abundant semigroups with strong Ehresmann transversals Math. Slovaca 68 3 549 562 10.1515/ms-2017-0124 Search in Google Scholar

Zhang, T. J., Tian, Z. J., Shi, Y. B. (2014). Good congruences on abundant semigroups with PSQ-adequate transversals. Semigroup Forum 89(2):462–468. ZhangT. J. TianZ. J. ShiY. B. 2014 Good congruences on abundant semigroups with PSQ-adequate transversals Semigroup Forum 89 2 462 468 10.1007/s00233-014-9584-1 Search in Google Scholar

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