Search

We shall prove a variety of results on certain types of Putcha semigroups such as: left weakly commutative semigroups, 𝓡-commutative semigroups, 𝓡𝓒-commutative semigroups, generalized conditionally commutative semigroups.

We prove first that every $$\mathcal {H}$$ H -commutative semigroup is stable. Using this result [and some results from the standard text (Nagy, Special classes of semigroups, Kluwer, Dordrecht, 2001)], we give two equivalent conditions for a semigroup to be an archimedean $$\mathcal {H}$$ H -commutative semigroup containing an idempotent element. It turns out that this result can be partially extended to $$\mathcal {L}$$ L -commutative semigroups and quasi-commutative semigroups.

We show without the Kuratowski-Zorn lemma that the set of all nilpotent elements of a medial semigroup (with zero) is the set-theoretic intersection of all its prime ideals. Moreover, some applications of the above theorem are given.

We study some special types of bands of E-inversive unipotent semigroups. It has been proved that in any R-semigroup S, which is a band of E-inversive unipotent semigroups, the set of its regular elements is a retract of S. Also, some characterizations of E-inversive rectangular bands of unipotent semigroups are given. This theorem extends nearly 40-old results from the theory of epigroups. In fact, a more general result is valid in some special subclass of the class of E-inversive semigroups; this result seems to be (partially) new for all epigroups.

A classical result of semigroup theory says that any finite congruence-free semigroup S (i.e., S has exactly two congruences) without zero such that card ( S ) > 2 is a simple group. We shall show that an analogous result holds for any infinite topologically congruence-free compact semigroup (a compact semigroup A is topologically congruence-free if the set of its algebraic congruences ρ for which A / ρ is a compact semigroup, is equal to { 1 A , A × A } ). In fact, every such semigroup must be a metric Lie group with cardinality c . Also, we prove that all topologically congruence-free compact semigroups with zero are (unfortunately) finite.

We prove that any permutative k-exponential epigroup is a normal band of unipotent epigroups. Moreover, we describe the least regular congruence on such semigroups. Finally, we give an example of a...

We show that an E-inversive semigroup S has a completely simple kernel KS if and only if it contains a primitive idempotent (in that case, KS is the set-theoretic union of the groups eSe, where e is a primitive idempotent of S). Along the way, some equivalent conditions for a semigroup to be E-inversive are given. Moreover, some applications of the above theorem will be pointed out.

A semigroup is called eventually regular if each of its elements has a regular power. In this paper we study certain fundamental congruences on an eventually regular semigroup. We generalize some results of Howie and Lallement (1966) and LaTorre (1983). In particular, we give a full description of the semilattice of group congruences (together with the least such a congruence) on an arbitrary eventually regular (orthodox) semigroup. Moreover, we investigate UBG-congruences on an eventually regular semigroup. Finally, we study the eventually regular subdirect products of an E-unitary semigroup and a Clifford semigroup.

In this paper we consider the following two equivalence relations on the lattice 𝒞(S) of all congruences on a semigroup S with the set of idempotents E S : where kerρ = {x ∈ S: (x, x 2) ∈ ρ} and trρ = ρ ∩ (E S × E S ) (ρ ∈ 𝒞(S)). We show that if S is idempotent-surjective, then every κ-class ρκ (ρ ∈ 𝒞(S)) coincides with some interval [ρκ, ρκ]. We study idempotent pure and E-disjunctive congruences on an arbitrary idempotent-surjective semigroup. In particular, we show that if τ is the greatest idempotent pure congruence on an idempotent-surjective semigroup S, then S/τ is E-disjunctive (see Section 2). In Section 3 we investigate the trace class ρθ (ρ ∈ 𝒞(S)) of an arbitrary idempotent-surjective semigroup S. Also, we study the connections between ρ ∈ 𝒞(S) and four associated (with ρ) congruences ρκ, ρκ, ρθ, ρθ. In particular, using a conjunction between the least Clifford congruence ξ (on an eventually regular semigroup) and the congruences ξκ, ξθ, we give the necessary and sufficient conditions for ξ t...

A semigroup $S$ is called idempotent-surjective (respectively, regular-surjective) if whenever $\rho $ is a congruence on $S$ and $a\rho $ is idempotent (respectively, regular) in $S/ \rho $, then there is $e\in {E}_{S} \cap a\rho $ (respectively, $r\in \mathrm{Reg} (S)\cap a\rho $), where ${E}_{S} $ (respectively, $\mathrm{Reg} (S)$) denotes the set of all idempotents (respectively, regular elements) of $S$. Moreover, a semigroup $S$ is said to be idempotent-regular-surjective if it is both idempotent-surjective and regular-surjective. We show that any regular congruence on an idempotent-regular-surjective (respectively, regular-surjective) semigroup is uniquely determined by its kernel and trace (respectively, the set of equivalence classes containing idempotents). Finally, we prove that all structurally regular semigroups are idempotent-regular-surjective.

We show that there is an inclusion-preserving bijection between the set of all normal subsemigroups of a semigroup S and the set of all group congruences on S. We describe also group congruences on E-inversive (E-)semigroups. In particular, we generalize the result of Meakin (J. Aust. Math. Soc. 13:259–266, 1972) concerning the description of the least group congruence on an orthodox semigroup, the result of Howie (Proc. Edinb. Math. Soc. 14:71–79, 1964) concerning the description of ρ∨σ in an inverse semigroup S, where ρ is a congruence and σ is the least group congruence on S, some results of Jones (Semigroup Forum 30:1–16, 1984) and some results contained in the book of Petrich (Inverse Semigroups, 1984). Also, one of the main aims of this paper is to study of group congruences on E-unitary semigroups. In particular, we prove that in any E-inversive semigroup, \(\mathcal{H}\cap\sigma\subseteq\kappa\), where κ is the least E-unitary congruence. This result is equivalent to the statement that in an arbitrary E-unitary E-inversive semigroup S, \(\mathcal{H}\cap\sigma= 1_{S}\).

A semigroup S is called η-simple if S has no semilattice congruences except S×S. Tamura in (Semigroup Forum 24:77–82, 1982) studied η-simple semigroups with a unique idempotent. In the present paper we consider a more general situation, that is, we investigate η-simple semigroups (without zero) with a least idempotent. Moreover, we study η∗-simple semigroups with zero which contain a least non-zero idempotent.

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.

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.