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 _{V}

#### Keywords

- weakly U-abundant semigroup
- Ehresmann transversal
- congruence
- equivalence triple

#### MSC 2010

- 20M10

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.

Let

For any _{U}_{U}_{U}_{U}_{U}^{*} and ^{+}, respectively.

For a semigroup denoted by another letter, say _{U}

_{U} be an E-semigroup. Then

^{*} = (^{*} ^{*} ^{+} = (^{+})^{+}

Let _{V}_{U}_{V}_{U}_{V} a_{U}

_{V} is a ∼ -w-V-a subsemigroup of a w-U-a semigroup S_{U}, then_{U} satisfies c-c, then, so dose T_{V}.

Call a-w-V-a subsemigroup _{V}_{U}_{V}_{U}^{+}_{x}_{x}_{x}_{x}

_{U} be a w-U-a semigroup with (C) and an E-transversal T_{V}. Then

_{x}^{+} _{x}^{*};

_{e}_{f}

A subset _{U}

_{V} be an E-transversal of a w-U-a semigroup S_{U}. Then T_{V} is a q-ideal of S_{U} if and only if T_{V} is a b-ideal of S_{U}.

Suppose that _{V}_{V}_{U}_{V}_{U}_{V}_{U}

We call an E-transversal _{V}_{U}_{V}_{U}_{V}_{U} strong

Define _{V}.

_{V} be a s-transversal of a w-U-a semigroup S_{U} with (C). Then:

_{U}

_{U}

Let

A (

(B1) There exist two mappings

(B2) Mappings ⊗:

(B3) For any

In the rest of this paper, we use

_{V} is an E-semigroup in which V is an order ideal of E_{f,e} by [f, e])

_{E}is a w-E-a semigroup with (C) and an s-transversal isomorphic to T

_{V}. Conversely, every w-U-a semigroup S

_{U}with (C) and an s-transversal T

_{V}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].

In this section, we describe the g-congruences on a w-U-a semigroup _{U}_{V}

A congruence _{U}^{+},^{+}) ∈ ^{*}, y^{*}^{I}^{Λ} be n-equivalences on ^{T}^{I}^{T}^{Λ}) an

(C.1) ^{I}|_{V}^{Λ}_{V}^{T} |_{V}

(C.2) (∀ ^{I}g^{T}

(C.3) (∀^{Λ}^{T}

Define a relation ^{(ρI,ρT,ρΛ)} on the semigroup

_{U} with (C) and a s-transversal T_{V} as in Lemma 7, and (ρ^{I}, ρ^{T}, ρ^{Λ}^{(ρI,ρT,ρΛ)}

For any e-triple (^{I},ρ^{T},ρ^{Λ}) ∈ ^{(ρI,ρT,ρΛ)} is an equivalence. Supposing that (^{(ρI,ρT,ρΛ)} (^{T} y^{I}g^{Λ}^{T}^{I}g^{T} y^{T} n^{T}^{+}^{T} n^{+} and so, by (C.1), we get ^{+}^{I}n^{+}. Therefore, (^{+})^{I}^{+}). Similarly, we can verify (^{*} * f^{Λ}(^{*} * h^{(ρI,ρT,ρΛ)} (^{(ρI,ρT,ρΛ)} is a congruence.

Supposing that (^{+},^{*}^{+},^{*}^{+},^{*}^{(ρI,ρT,ρΛ)} (^{+},^{*}^{+}^{T} y^{+} since ^{T}^{+}^{Λ}^{+}. Thus, we have the following relation
^{(ρI,ρT,ρΛ)} is a g-congruence.

Conversely, let

(E1) (∀^{+},^{+})_{I}g^{+},^{+})^{+},^{+});

(E2) (∀ ^{*},hψ^{*}_{Λ}^{*}, x^{*}, f^{*}, y^{*}, h

(E3) (∀_{T} y^{+}, ^{*}^{+},^{*}

We have demonstrated that _{I}_{Λ} are equivalences on _{T}

For any _{I}k^{+},^{+}) (^{+},^{+})(_{I}_{I}_{Λ} it is also an n-equivalence.

Further we have the cases 1–3:

Case 1. Obviously, _{I}|_{V}_{Λ}_{V}_{T} |_{V}

Case 2. For any ^{+}, ^{+} and _{I}g^{+},^{+})^{+},^{+}). So, for any (^{*}, z^{*}, j

That is to say,
^{*}^{+}_{S◦} ^{*}^{+}. Moreover, by (B3) and (B4), we have ^{*}^{+} = [^{*}^{+} = [

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

In conclusion, we have that (_{I}_{T}_{Λ}) is an e-triple on

By the direct part, ^{(ρI,ρT,ρΛ)} is a g-congruence. On the other hand, for any (^{(ρI,ρT,ρΛ)} (^{(ρI,ρT,ρΛ)} ⊆ ^{(ρI,ρT,ρΛ)}, and thus we have ^{(ρI,ρT,ρΛ)} =

Next, we give an example of a g-congruence on a w-U-a semigroup _{U}

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

It is easy to check that _{U′}_{U′}

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

Let

^{+} and ^{+}. By the proof of Theorem 1, we have

Define relation ≤ on

Then

_{ρ}^{I}, ρ^{T}, ρ^{Λ},

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

For any ^{+}, ^{+} and
_{i}_{i}_{i},x_{i}, f_{i}_{1}_{1}_{2}_{2} ···_{n}_{−1}_{n}j

Next, assume that
_{i}_{1}_{1}_{2}_{2} ···_{n}_{−1}_{n}_{−1}_{n}t

To sum up, we obtain the following theorem.

#### 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 |

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