On right congruences associated with the input semigroupS of automata

LetA=(M, S, δ) be an automaton without outputs whereM is a nonemptyset andS is a nonempty semigroup. Then the right congruences μM and μ _m associated withS have been expressed in many different ways (μ _M is called the Myhill-Nerode congruence onS). Also, their algebraic properties have been investigated. We have introduced the right congruences μ _S and μ_α onM and we have obtained necessary and sufficient conditions thatS/μ andM/w have nontrivialS-homomorphisms where μ andw are any right congruences onS andM respectively. The faithfulness ofS has been introduced.     

关于左型A半群上的fuzzy同余

引入了左富足半群上fuzzy右好同余和fuzzy右消去同余的概念,给出了左富足半群上fuzzy右好同余的性质和特征.在此基础上,给出了左型A半群上fuzzy右好同余和fuzzy右消去同余的性质.得到了左型A半群上的fuzzy右好同余为fuzzy右消去同余的充要条件.     

Rectangular group congruences on a semigroup

We study rectangular group congruences on an arbitrary semigroup. Some of our results are an extension of the results obtained by Masat (Proc. Am. Math. Soc. 50:107–114, 1975). We show that each rectangular group congruence on a semigroup S is the intersection of a group congruence and a matrix congruence and vice versa, and this expression is unique, when S is E-inversive. Finally, we prove that every rectangular group congruence on an E-inversive semigroup is uniquely determined by its kernel and trace.     

The direct product of right congruences

In a number of earlier papers the study of the structure of semigroups has been approached by means of right congruences. Such an approach seems appropriate since a right congruence is one of the possible analogs of both the right ideal of a ring and the subgroup in a group. Each of these substructures plays a strong role in the study of the structure of their respective systems. In both the ring and the group the internal direct product is naturally, and effectively, defined. However, what such an internal direct product should be for two right congruences of a semigroup is not so clear. In this paper we will offer a possible definition and consider some of the consequences of it. We will also extend some of these results to automata.     

Certain congruences on E-inversive E-semigroups

A semigroup S is called E-inversive if for every a ∈ S there exists x ∈ S such that ax is idempotent. S is called E-semigroup if the set of idempotents of S forms a subsemigroup. In this paper some special congruences on E-inversive E-semigroups are investigated, such as the least group congruence, a certain semilattice congruence, some regular congruences and a certain idempotent-separating congruence.     

A problem on bands

In this paper, we consider an open problem proposed by Petrich and Reilly: What are necessary and sufficient conditions on a completely regular semigroup S in order that the trace relation T on the lattice of congruences on S is equal to the identity relation? By constructing some special congruences on S, we prove that T=ε if and only if S is a band.     

Semigroups whose lattices of right congruences are semiatomic

The main theorems presented here are characterizations of a semigroup with a left identity whose lattice of right congruences is semiatomic. These theorems are preceded by a number of results on minimal right congruences.     

Left Reductive Congruences on Semigroups

A semigroup S is called a left reductive semigroup if, for all elements a,b∈S, the assumption “xa=xb for all x∈S” implies a=b. A congruence α on a semigroup S is called a left reductive congruence if the factor semigroup S/α is left reductive. In this paper we deal with the left reductive congruences on semigroups. Let S be a semigroup and ? a congruence on S. Consider the sequence ? ⁽⁰⁾?? ⁽¹⁾???? ⁽ⁿ⁾?? of congruences on S, where ? ⁽⁰⁾=? and, for an arbitrary non-negative integer n, ? ⁽ⁿ⁺¹⁾ is defined by (a,b)∈? ⁽ⁿ⁺¹⁾ if and only if (xa,xb)∈? ⁽ⁿ⁾ for all x∈S. We show that $\bigcup_{i=0}^{\infty}\varrho^{(i)}\subseteq \mathit{lrc}(\varrho )$ for an arbitrary congruence ? on a semigroup S, where lrc(?) denotes the least left reductive congruence on S containing ?. We focuse our attention on congruences ? on semigroups S for which the congruence $\bigcup_{i=0}^{\infty}\varrho^{(i)}$ is left reductive. We prove that, for a congruence ? on a semigroup S, $\bigcup_{i=0}^{\infty}\varrho^{(i)}$ is a left reductive congruence of S if and only if $\bigcup_{i=0}^{\infty}\iota_{(S/\varrho)}^{(i)}$ is a left reductive congruence on the factor semigroup S/? (here ι _(S/?) denotes the identity relation on S/?). After proving some other results, we show that if S is a Noetherian semigroup (which means that the lattice of all congruences on S satisfies the ascending chain condition) or a semigroup in which S ⁿ =S ⁿ⁺¹ is satisfied for some positive integer n then the universal relation on S is the only left reductive congruence on S if and only if S is an ideal extension of a left zero semigroup by a nilpotent semigroup. In particular, S is a commutative Noetherian semigroup in which the universal relation on S is the only left reductive congruence on S if and only if S is a finite commutative nilpotent semigroup.     

Ordered groupoid quotients and congruences on inverse semigroups

We introduce a preorder on an inverse semigroup S associated to any normal inverse subsemigroup N, that lies between the natural partial order and Green’s \({\mathcal {J}}\)–relation. The corresponding equivalence relation \(\simeq _N\) is not necessarily a congruence on S, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on S. We give an application to the congruence and certain normal inverse subsemigroups associate to an inverse monoid presentation.     

Using filters to describe congruences and band congruences of semigroups

It is well known that the smallest semilattice congruence can be described via filters. We generalise this result to the smallest left (right) normal band congruences and also to arbitrary semilattice (left normal band, right normal band) congruences, describing them all via filters. To achieve this, we introduce filters relative to arbitrary quasiorders on a semigroup (traditional filters are filters relative to the smallest negative operation-compatible quasiorder). We study congruences which can be described via filters. We show that the lattice of semilattice (left normal band, right normal band) congruences is a homomorphic image of the lattice of negative (right negative, left negative) operation-compatible quasiorders.     

Algebras whose equivalence relations are congruences

It is proved that all the equivalence relations of a universal algebra A are its congruences if and only if either |A| ≤ 2 or every operation f of the signature is a constant (i.e., f(a ₁ , . . . , a _n ) = c for some c ∈ A and all the a ₁ , . . . , a _n ∈ A) or a projection (i.e., f(a ₁ , . . . , a _n ) = a _i for some i and all the a ₁ , . . . , a _n ∈ A). All the equivalence relations of a groupoid G are its right congruences if and only if either |G| ≤ 2 or every element a ∈ G is a right unit or a generalized right zero (i.e., x _a  = y _a for all x, y ∈ G). All the equivalence relations of a semigroup S are right congruences if and only if either |S| ≤ 2 or S can be represented as S = A∪B, where A is an inflation of a right zero semigroup, and B is the empty set or a left zero semigroup, and ab = a, ba = a ² for a ∈ A, b ∈ B. If G is a groupoid of 4 or more elements and all the equivalence relations of it are right or left congruences, then either all the equivalence relations of the groupoid G are left congruences, or all of them are right congruences. A similar assertion for semigroups is valid without the restriction on the number of elements.     

Representation of congruences on regular semigroups with inverse transversals

For a regular semigroup with an inverse transversal, we have Saito’s structureW(I,S ^o, Λ, *, {α, β}). We represent congruences on this kind of semigroups by the so-called congruence assemblage which consist of congruences on the structure component partsI,S ^o and Λ. The structure of images of this type of semigroups is also presented. This work is supported by Natural Science Foundation of Guangdong Province     

On congruences on \mathcal{T} -regular semigroups

A nontrivial regular semigroup S with zero in which every interval of idempotents is a finite chain is said to be -regular. The structure of these semigroups is described in terms of trees of completely 0-simple semigroups. For S in this form, we study congruences which we express in terms of congruence aggregates. We determine the inclusion relation, meet and join of congruences, their kernel and trace, and the ends of the intervals which form their classes. We characterize those S for which the kernel relation on the congruence lattice is a congruence, and those for which the operators and are homomorphisms.     

含幺Clifford半群上的Rees矩阵半群的同余和正规加密群结构

给出了含幺Clifford半群上的Rees矩阵半群S的正规加密群结构,证明了在含幺Clifford半群上的Rees矩阵半群S上以下两个条件是等价的:(1)S上的同余ρ是完全单半群同余;(2)S上的同余ρ和S上的相容组之间存在保序双射.最后还证明了S上的完全单半群同余所构成的同余格是半模的.     

Varieties with equationally definable factor congruences

A variety has definable factor congruences if there is a first order formula φ which defines each factor congruence in terms of its associated central element. We study the case in which φ is a conjunction of equations.     

Ascending chain conditions on principal left and right ideals for semidirect products of semigroups

In this paper we investigate the ascending chain conditions on principal left and right ideals for semidirect products of semigroups and show how this is connected to the corresponding problem for rings of skew generalized power series. Let S be a left cancellative semigroup with a unique idempotent e, T a right cancellative semigroup with an idempotent f and $\omega: T \to \operatorname {End}(S)$ a semigroup homomorphism such that ??(f)=id _S . We show that in this case the semidirect product S? _?? T satisfies the ascending chain condition for principal left ideals (resp. right ideals) if and only if S and T satisfy the ascending chain condition for principal left ideals (resp. right ideals and $\operatorname {Im}\omega(t)$ is closed for complete inverses for all t??T). We also give several examples to show that for more general semigroups these implications may not hold.     

Completely inverse AG ∗∗-groupoids

A completely inverse AG ^??-groupoid is a groupoid satisfying the identities (xy)z=(zy)x, x(yz)=y(xz) and xx ^?1=x ^?1 x, where x ^?1 is a unique inverse of x, that is, x=(xx ^?1)x and x ^?1=(x ^?1 x)x ^?1. First we study some fundamental properties of such groupoids. Then we determine certain fundamental congruences on a completely inverse AG ^??-groupoid; namely: the maximum idempotent-separating congruence, the least AG-group congruence and the least E-unitary congruence. Finally, we investigate the complete lattice of congruences of a completely inverse AG ^??-groupoids. In particular, we describe congruences on completely inverse AG ^??-groupoids by their kernel and trace.     

Certain congruences on free trioids

Loday and Ronco introduced the notion of a trioid and constructed the free trioid of rank 1. This paper is devoted to the study of congruences on trioids. We characterize the least dimonoid congruences and the least semigroup congruence on the free (commutative, rectangular) trioid.