The non-nilpotent graph of a semigroup

We associate a graph ${\mathcal{N}}_{S}$ with a semigroup S (called the upper non-nilpotent graph of S). The vertices of this graph are the elements of S and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Malcev). In case S is a group this graph has been introduced by A. Abdollahi and M.?Zarrin and some remarkable properties have been proved. The aim of this paper is to study this graph (and some related graphs, such as the non-commuting graph) and to discover the algebraic structure of S determined by the associated graph. It is shown that if a finite semigroup S has empty upper non-nilpotent graph then S is positively Engel. On the other hand, a semigroup has a complete upper non-nilpotent graph if and only if it is a completely simple semigroup that is a band. One of the main results states that if all connected ${\mathcal{N}}_{S}$ -components of a semigroup S are complete (with at least two elements) then S is a band that is a semilattice of its connected components and, moreover, S is an iterated total ideal extension of its connected components. We also show that some graphs, such as a cycle C _n on n vertices (with n??5), are not the upper non-nilpotent graph of a semigroup. Also, there is precisely one graph on 4 vertices that is not the upper non-nilpotent graph of a semigroup with 4 elements. This work also is a continuation of earlier work by Okni??ski, Riley and the first named author on (Malcev) nilpotent semigroups.     

On semirings whose additive reduct is a semilattice

A semiring S whose additive reduct is a semilattice is called a k-regular semiring if for every a∈S there is x∈S such that a+axa=axa. For a semigroup F, the power semiring P(F) is a k-regular semiring if and only if F is a regular semigroup. An element e∈S is a k-idempotent if e+e ²=e ². Basic properties of k-regular semirings whose k-idempotents are commutative have been studied.     

The value semiring of an algebroid curve

Abstract

We study the value semiring Γ, equipped with the tropical operations, associated to an algebroid curve. As a set, Γ determines and is determined by the well-known value semigroup S and we prove that Γ is always finitely generated in contrast to S. In particular, for a plane curve, we present a straightforward way to obtain Γ in terms of the semiring (or the semigroup) of each branch of the curve and the mutual intersection multiplicity of its branches. In the analytic case, this allows us to relate the results of Zariski and Waldi that characterize the topological type of the curve.     

Idempotent semirings with a commutative additive reduct

For every semigroup S , we define a congruence relation ρ on the power semiring (P(S),\cup,\circ) of S . If S is a band, then P(S)/ρ is an idempotent semiring . This enables us to find models for the free objects in the variety of idempotent semiring s whose additive reduct is a semilattice. December 28, 1999     

On the subsemiring generated by almost idempotents of a k-regular semiring

An element e of a semiring S with commutative addition is called an almost idempotent if \(e + e^2 = e^2\). Here we characterize the subsemiring \(\langle E(S)\rangle \) generated by the set E(S) of all almost idempotents of a k-regular semiring S with a semilattice additive reduct. If S is a k-regular semiring then \(\langle E(S)\rangle \) is also k-regular. A similar result holds for the completely k-regular semirings, too.     

ON THE STRUCTURE OF REGULAR CRYPTO SEMIGROUPS

Superabundant semigroups are generalizations of completely regular semigroups written the class of abundant semigroups. It has been shown by Fountain that an abundant semigroup is superabundant if and only if it is a semilattice of completely J ^*-simple semigroups. Reilly and Petrich called a semigroup S cryptic if the Green's relation H is a congruence on S. In this paper, we call a superabundant semigroup S a regular crypto semigroup if H ^* is a congruence on S such that S/H ^* is a regular band. It will be proved that a superabundant semigroup S is a regular crypto semigroup if and only if S is a refined semilattice of completely J ^*-simple semigroups. Thus, regular crypto semigroups are generalization of the cryptic semigroups as well as abundant semigroups.     

Inequalities for Gondran-Minoux rank and idempotent semirings

The rank-sum, rank-product, and rank-union inequalities for Gondran-Minoux rank of matrices over idempotent semirings are considered. We prove these inequalities for matrices over quasi-selective semirings without zero divisors, which include matrices over the max-plus semiring. Moreover, it is shown that the inequalities provide the linear algebraic characterization for the class of quasi-selective semirings. Namely, it is proven that the inequalities hold for matrices over an idempotent semiring S without zero divisors if and only if S is quasi-selective. For any idempotent semiring which is not quasi-selective it is shown that the rank-sum, rank-product, and rank-union inequalities do not hold in general. Also, we provide an example of a selective semiring with zero divisors such that the rank-sum, rank-product, and rank-union inequalities do not hold in general.     

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.     

On disjunctions of algebraic sets in completely simple semigroups

A semigroup S is called an equational domain if any finite union of algebraic sets over S is algebraic. We give some necessary and su?cient conditions for a completely simple semigroup to be an equational domain.     

On semirings satisfying the Baer criterion

It is well-known that for modules over rings the Baer injectivity criterion takes place. In this paper we prove that under one additional condition this criterion is also valid for modules over semirings. We prove that a semiring S satisfies the Baer criterion if and only if all injective (with respect to one-sided ideals of S) semimodules satisfy the above condition. We propose a newmethod for constructing semirings satisfying the Baer criterion.     

Semigroup actions on ordered groupoids

In this paper we prove that if S is a commutative semigroup acting on an ordered groupoid G, then there exists a commutative semigroup S? acting on the ordered groupoid G?:=(G × S)/ρ? in such a way that G is embedded in G?. Moreover, we prove that if a commutative semigroup S acts on an ordered groupoid G, and a commutative semigroup S? acts on an ordered groupoid G? in such a way that G is embedded in S?, then the ordered groupoid G? can be also embedded in G?. We denote by ρ? the equivalence relation on G × S which is the intersection of the quasi-order ρ (on G × S) and its inverse ρ ^?1.     

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.     

Ω-compact semigroups having congruence extension property

The congruence extension property (CEP) of semigroups has been extensively studied by a number of authors. We call a compact semigroup S an Ω-compact semigroup if the set of all regular elements of S forms an ideal of S. In this note, we characterize the Ω-compact semigroup having (CEP). Our result extends a recent result obtained by X.J. Guo on the congruence extension property of strong Ω-compact semigroups which is a semigroup containing precisely one regular D-class.     

On the applicability to semirings of two theorems from the theory of rings and modules

Problems concerning the extension of the Baer criterion for injectivity and embedding theorem of an arbitrary module over a ring into an injective module to the case of semirings are treated. It is proved that a semiring S satisfies the Baer criterion and every S-semimodule can be embedded in an injective semimodule if and only if S is a ring.     

On left regular and intra-regular ordered semigroups

We study the decomposition of left regular ordered semigroups into left regular components and the decomposition of intra-regular ordered semigroups into simple or intra-regular components, adding some additional information to the results considered in [KEHAYOPULU, N.: On left regular ordered semigroups, Math. Japon. 35 (1990), 1057–1060] and [KEHAYOPULU, N.: On intra-regular ordered semigroups, Semigroup Forum 46 (1993), 271–278]. We prove that an ordered semigroup S is left regular if and only if it is a semilattice (or a complete semilattice) of left regular semigroups, equivalently, it is a union of left regular subsemigroups of S. Moreover, S is left regular if and only if it is a union of pairwise disjoint left regular subsemigroups of S. The right analog also holds. The same result is true if we replace the words “left regular” by “intraregular”. Moreover, an ordered semigroup is intra-regular if and only if it is a semilattice (or a complete semilattice) of simple semigroups. On the other hand, if an ordered semigroup is a semilattice (or a complete semilattice) of left simple semigroups, then it is left regular, but the converse statement does not hold in general. Illustrative examples are given.     

Orders in Strict Regular Semigroups

 A subsemigroup S of a semigroup Q is an order in Q if for every there exist such that , where a and d are contained in (maximal) subgroups of Q, and and are their inverses in these subgroups. A regular semigroup S is strict if it is a subdirect product of completely (0-)simple semigroups. We construct all orders and involutions in Auinger’s model of a strict regular semigroup. This is used to find necessary and sufficient conditions on an involution on an order S in a strict regular semigroup Q for extendibility to an involution on Q.     

Direct sums of injective semimodules and direct products of projective semimodules over semirings

We prove that if the direct sum of a family of semimodules over a semiring S is an injective semimodule or if the direct product of a family of semimodules over S is a projective semimodule, then the cardinality of the subfamily consisting of all semimodules which are not modules is strictly less than the cardinality of S. As a consequence, we obtain semiring analogs of well-known characterizations of classical semisimple, quasi-Frobenius, and one-sided Noetherian rings.     

Weakly separative weakly commutative semigroups

A semigroup S is called a weakly commutative semigroup if, for every a,b∈S, there is a positive integer n such that (ab) ⁿ ∈Sa∩bS. A semigroup S is called archimedean if, for every a,b∈S, there are positive integers m and n such that a ⁿ ∈SbS and b ^m ∈SaS. It is known that every weakly commutative semigroup is a semilattice of weakly commutative archimedean semigroups. A semigroup S is called a weakly separative semigroup if, for every a,b∈S, the assumption a ²=ab=b ² implies a=b. In this paper we show that a weakly commutative semigroup is weakly separative if and only if its archimedean components are weakly cancellative. This result is a generalization of Theorem 4.16 of Clifford and Preston (The Algebraic Theory of Semigroups, Am. Math. Soc., Providence, 1961).