We consider semigroups such that the universal left congruence ω is finitely generated. Certainly a left noetherian semigroup, that is, one in which all left congruences are finitely generated, satisfies our condition. In the case of a monoid the condition that ω is finitely generated is equivalent to a number of pre-existing notions. In particular, a monoid satisfies the homological finiteness condition of being of type left-FP 1 exactly when ω is finitely generated.Our investigations enable us to classify those semigroups such that ω is finitely generated that lie in certain important classes, such as strong semilattices of semigroups, inverse semigroups, Rees matrix semigroups (over semigroups) and completely regular semigroups. We consider closure properties for the class of semigroups such that ω is finitely generated, including under morphic image, direct product, semi-direct product, free product and 0-direct union. Our work was inspired by the stronger condition, stated for monoids in the work of White, of being pseudo-finite. Where appropriate, we specialise our investigations to pseudo-finite semigroups and monoids. In particular, we answer a question of Dales and White concerning the nature of pseudo-finite monoids.Lemma 2.5 Let S be a semigroup and let ω S be finitely generated by H ⊆ S 2 . Suppose ω S = K for some K ⊆ S 2 . Then there exists a finite subset K of K such that ω S = K .Further, if there exists m ∈ N such that for any a, b ∈ S, there is an H -sequence from a to b of length at most m, then there is an m ∈ N such that for any a, b ∈ S, there is a K -sequence from a to b of length at most m .Proof The first statement is well known, but we give a short proof here for completeness and convenience.We are given that ω S = H = K . Let (h, k) ∈ H . Then there is a K -sequence of length n := n(h, k) h = t 1 c 1 , t 1 d 1 = t 2 c 2 , . . . , t n d n = k,

Direct, semidirect and Zappa-Szép products provide tools to decompose algebraic structures, with each being a natural generalisation of its predecessor. In this paper we examine Zappa-Szép products of monoids and semigroups and investigate generalised Greens relations R * , L * , R E and L E for these Zappa-Szép products. We consider a left restriction semigroup S with semilattice of projections E and define left and right actions of S on E and E on S, respectively, to form the Zappa-Szép product E S. We further investigate properties of E S and show that S is a retract of E S. We also find a subset T of E S which is left restriction.

This paper concerns a class of semigroups that arise as products U S, associated to what we call 'action pairs'. Here U and S are subsemigroups of a common monoid and, roughly speaking, S has an action on the monoid completion U 1 that is suitably compatible with the product in the over-monoid.The semigroups encapsulated by the action pair construction include many natural classes such as inverse semigroups and (left) restriction semigroups, as well as many important concrete examples such as transformational wreath products, linear monoids, (partial) endomorphism monoids of independence algebras, and the singular ideals of many of these. Action pairs provide a unified framework for systematically studying such semigroups, within which we build a suite of tools to ensure a comprehensive understanding of them. We then apply our abstract results to many special cases of interest.The first part of the paper constitutes a detailed structural analysis of semigroups arising from action pairs. We show that any such semigroup U S is a quotient of a semidirect product U ⋊S, and we classify all congruences on semidirect products that correspond to action pairs. We also prove several covering and embedding theorems, each of which naturally extends celebrated results of McAlister on proper (a.k.a. E-unitary) inverse semigroups.The second part of the paper concerns presentations by generators and relations for semigroups arising from action pairs. We develop a substantial body of general results and techniques that allow us to build presentations for U S out of presentations for the constituents U and S in many cases, and then apply these to several examples, including those listed above. Due to the broad applicability of the action pair construction, many results in the literature are special cases of our more general ones.

The aim of this paper is to study λ-semidirect and λ-Zappa-Szép products of restriction semigroups. The former concept was introduced for inverse semigroups by Billhardt, and has been extended to some classes of left restriction semigroups. The latter was introduced, again in the inverse case, by Gilbert and Wazzan. We unify these concepts by considering what we name the scaffold of a Zappa-Szép product S T where S and T are restriction. Under certain conditions this scaffold becomes a category. If one action is trivial, or if S is a semilattice and T a monoid, the scaffold may be ordered so that it becomes an inductive category. A standard technique, developed by Lawson and based on the Ehresmann-ScheinNambooripad result for inverse semigroups, allows us to define a product on our category. We thus obtain restriction semigroups that are λ-semidirect products and λ-Zappa-Szép products, extending the work of Billhardt and of Gilbert and Wazzan. Finally, we explicate the internal structure of λ-semidirect products.