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LetSbe any semigroup anda,s ∈ S. Ifa = asa, thensis anassociateofa. A subgroupGofSis anassociate subgroupofSif everya ∈ Shas a unique associatea* inG. It turns out thatG = Hzfor some idempotentz, thezenithofS. The mappinga → a* is a unary operation onS. We say thatSismonogenicifSis generated, as a unary semigroup, by a single element. We embark upon the problem of the structure of monogenic semigroups in this sense by characterizing monogenic ones belonging to completely simple semigroups, normal cryptogroups, orthogroups, combinatorial semigroups, cryptic medial semigroups, cryptic orthodox semigroups, and orthodox monoids. In each of these cases, except one, we construct a free object. The general problem remains open. [ABSTRACT FROM AUTHOR]

We consider unipotent, weakly cancellative, left cancellative, right cancellative, cancellative monoids and groups with the unary operation of mapping onto the identity element and bands with the identity mapping as unary operations. Then form Malcev products of each of the former with varieties of rectangular bands, semilattices, normal bands and all bands. Regarding the resulting semigroups with the induced unary operations, we characterize these classes, some of them we provide with a construction, and determine a basis for their implications, for they all turn out to be quasivarieties. We also determine their relationship which we clarify by an inclusion diagram. [ABSTRACT FROM AUTHOR]

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ be a semigroup. Elements $a,b$ of $S$ are $\widetilde{\mathscr{R}}$-related if they have the same idempotent left identities. Then $S$ is weakly left ample if (1) idempotents of $S$ commute, (2) $\widetilde{\mathscr{R}}$ is a left congruence, (3) for any $a \in S$, $a$ is $\widetilde{\mathscr{R}}$-related to a (unique) idempotent, say $a^+$, and (4) for any element $a$ and idempotent $e$ of $S$, $ae=(ae)^+a$. Elements $a,b$ of $S$ are $\mathscr{R}^*$-related if, for any $x,y \in S^1$, $xa=ya$ if and only if $xb=yb$. Then $S$ is left ample if it satisfies (1), (3) and (4) relative to $\mathscr{R}^*$ instead of $\widetilde{\mathscr{R}}$. Further, $S$ is (weakly) ample if it is both (weakly) left and right ample. We establish several characterizations of these classes of semigroups. For weakly left ample ones we provide a construction of all such semigroups with zero all of whose nonzero idempotents are primitive. Among characterizations of weakly ample semigroups figure (strong) semilattices of unipotent monoids, and among those for ample semigroups, (strong) semilattices of cancellative monoids. This describes the structure of these two classes of semigroups in an optimal way, while, for the ‘one-sided’ case, the problem of structure remains open. [ABSTRACT FROM PUBLISHER]

Let $\mathcal{U}$ be the class of all unipotent monoids and $\mathcal{B}$ the variety of all bands. We characterize the Malcev product $\mathcal{U} \circ \mathcal{V}$ where $\mathcal{V}$ is a subvariety of $\mathcal{B}$ low in its lattice of subvarieties, $\mathcal{B}$ itself and the subquasivariety $\mathcal{S} \circ \mathcal{RB}$, where $\mathcal{S}$ stands for semilattices and $\mathcal{RB}$ for rectangular bands, in several ways including by a set of axioms. For members of some of them we describe the structure as well. This succeeds by using the relation $\widetilde{\mathcal{H}}= \widetilde{\mathcal{L}} \cap \widetilde{\mathcal{R}}$, where $a\;\,\widetilde{\mathcal{L}}\;\,b$ if and only if a and b have the same idempotent right identities, and $\widetilde{\mathcal{R}}$ is its dual. We also consider $(\mathcal{U} \circ \mathcal{RB}) \circ \mathcal{S}$ which provides the motivation for this study since $(\mathcal{G} \circ \mathcal{RB}) \circ \mathcal{S}$ coincides with completely regular semigroups, where $\mathcal{G}$ is the variety of all groups. All this amounts to a generalization of the latter: $\mathcal{U}$ instead of $\mathcal{G}$. [ABSTRACT FROM AUTHOR]

An orthogroup is a completely regular semigroup whose idempotents form a subsemigroup. For any semigroup S and a, x ∈ S, x is an associate of a if a = axa. A subgroup G of S is an associate subgroup of S if for every element a of S, G contains exactly one associate of a. An idempotent e of S is medial if for any element s of S which is a product of idempotents holds s = ses. We characterize orthogroups with an associate subgroup in several ways using the structural description of orthogroups as a semilattice of rectangular groups. Using the Blyth-Martins structure theorem for semigroups with an associate sub-group whose identity element is medial, we provide multiple characterizations of orthogroups, and some related semigroups, with an associate subgroup. [ABSTRACT FROM AUTHOR]