We show that every such semigroup is a homomorphic image of a subsemigroup of some finite inverse semigroup. This shows that the pseudovariety generated by the finite inverse semigroups consists of exactly the finite semigroups with commuting idempotents.

An inverse semigroup S is said to be modular if its lattice 𝓛𝓕 (S) of inverse subsemigroups is modular. We show that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that such a semigroup S is modular if and only if (I) S is combinatorial, (II) its semilattice E of idempotents is “Archimedean” in S, (III) its maximum group homomorphic image G is locally cyclic and (IV) the poset of idempotents of each 𝓓-class of S is either a chain or contains exactly one pair of incomparable elements, each of which is maximal. Thus in view of earlier results of the second author a simple modular inverse semigroup is “almost” distributive. The bisimple modular inverse semigroups are explicitly constructed. It is remarkable that exactly one of these is nondistributive.

K. D. Magill has investigated the semigroup generated by the idempotent continuous mappings of a topological space into itself and examined whether this semigroup determines the space to within homeomorphism. By analogy with this (and related work of Bridget Bos Baird) we now consider the semigroup generated by nilpotent continuous partial mappings of a space into itself.

A semigroup is eventually regular if each of its elements has some power that is regular. Let 𝓚 be one of Green's relations and let ρ be a congruence on an eventually regular semigroup S. It is shown for 𝓚 = 𝓛, 𝓡 and 𝓓 that if A and B are regular elements of S/ρ that are 𝓚-related in S/ρ then there exist elements a ∈ A, b ∈ B such that a and b are 𝓚-related in S. The result is not true for 𝓗 or 𝓙.

Sandwich semigroups were introduced in [4], [5] and [6]. Green's relations (for regular elements) were characterized for these semigroups in [11] and [13]. Sandwich semigroups of continuous functions first made their appearance in [5]. In this paper, we consider only sandwich semigroups of continuous functions and we refer to them simply as sandwich semigroups. We now recall the definition. Let X and Y be topological spaces and fix a continuous function α from Y into X. Let S(X, Y, α) denote the semigroup of all continuous functions from X into Y where the product fg of f, g ε S(X, Y, α) is defined by fg = f ∘ α ∘ g. We refer to S(X, Y, α) as a sandwich semigroup with sandwich function α. If X = Y and α is the identity map then S(X, Y, α) is, of course, just S(X), the semigroup of all continuous selfmaps of X.

Whilst the Mal'cev product of completely regular varieties need not again be a variety, it is shown that in many important instances a variety is in fact obtained. However, unlike the product of group varieties this product is nonassociative.

Two important operators introduced by Reilly are studied in the context of Mal'cev products. These operators are shown to generate from any given variety one of the networks discovered by Pastijn and Trotter, enabling identities to be provided for the varieties in the network. In particular the join O V BG of the varieties of orthogroups and of bands of groups is determined, answering a question of Petrich.

The class of abundant semigroups originally arose from ‘homological’ considerations in the theory of S-systems: they are the semigroup theoretic counterparts of PP-rings. Cancellative monoids, full subsemigroups of regular semigroups as well as the multiplicative semigroups of PP-rings are abundant. In this paper we investigate the properties of Rees matrix semigroups over abundant semigroups. Some of our results generalise McAlister's work on regular Rees matrix semigroups.

We give a survey of some of the realisations that have been given of monogenic inverse semigroups and discuss their relation to one another. We then analyse the representations by bijections, combined under composition, of monogenic inverse semigroups, and classify these into isomorphism types. This provides a particularly easy way of classifying monogenic inverse semigroups into isomorphism types. Of interest is that we find two quite distinct representations by bijections of free monogenic inverse semigroups and show that all such representations must contain one of these two representations. We call a bijection of the form ai ↦ ai+1, i = 1,2,…, r − 1, a finite link of length r, and one of the form ai ↦ ai+1, i = 1,2…, a forward link. The inverse of a forward link we call a backward link. Two bijections u: A → B and r: C → D are said to be strongly disjoint if A ∩ C, A ∩ D, B ∩ C and B ∩ D are each empty. The two distinct representations of a free monogenic inverse semigroup, that we have just referred to, are first, such that its generator is the union of a counbtable set os finite links that are pairwise storongly disjoint part of any representation of a free monogenic inverse semigroup, the remaining part not affecting the isomorphism type. Each representation of a monogenic inverse semigroup that is not free contains a strongly disjoint part, determining it to within isomorphism, that is generated by either the strongly disjoint union of a finite link and a permutation or the strongly disjoint union of a finite and a forward link.

We characterize rings whose multiplicative subsemigroups containing 0 and the additive inverse of each element are subrings. In addition we consider commutative rings for which every non-constant multiplicative endormorphism that preserves additive inverses is a ring endomorphism, and we show that they belong to one of three easily-described classes of rings.

Four properties of congruences on a regular semigroup S are studied and compared. Let R, L and D denote Green's relations and let V = {(a, b) ∈ S × S|a and b are mutually inverse}. A congruence ρ on S is (1) rectangular provided ρ ∩ D = (ρ ∩ L) ° (ρ ∩ R), (2) V-commuting provided ρ ° V = V ° ρ, (3) (L, R)-commuting provided L ° ρ = ρ ° L, and R ° ρ = ρ ° R, and (4) idempotent-regular provided each idempotent ρ-class is a regular subsemigroup of S.

A rectangular congruence is (L, R)-commuting and a V-commuting congruence is idempotent-regular. If ρ is idempotent-regular and (L, R)-commuting then ρ is V-commuting. Examples and conditions are given to show what other implications among the four properties hold. In addition to characterizations of the properties, these are studied in the presence of other conditions on S. For example, if S is a stable regular semigroup, then each congruence under D is rectangular.

In this paper “a map” denotes an arbitrary (everywhere defined, or partial, or even multi-valued) mapping. A map is constant if any two elements belonging to its domain have precisely the same images under this map. We characterize those semigroups which can be isomorphic to semigroups of constant maps or to involuted semigroups of constant maps.

A regular semigroup S is said to be locally inverse if each local submonoid eSe, with e an idempotent, is an inverse semigroup. In this paper we apply known covering theorems for inverse semigroups and a covering theorem for locally inverse semigroups due to the author to obtain some covering theorems for locally inverse semigroups. The techniques developed here permit us to give an alternative proof for, and sligbt strengthening of, an important covering theorem for locally inverse semigroups due to F. Pastijn.

If CS(respectively, O) denotes the class of all completely simple semigroups (respectively, semigroups that are orthodox unions of groups) then CS(respectively, O) is a variety of algebras with respect to the operations of multiplication and inversion. The main result shows that the lattice of subvarieties of is a precisely determined subdirect product of the lattice of subvarieties of CSand the lattice of subvarieties of O. A basis of identities is obtained for any variety in terms of bases of identities for . Several operators on the lattice of subvarieties of are also introduced and studied.

A natural equivalence θ on the lattice of congruences λ(S) of a semigroup S is studied. For any eventually regular semigroup S, it is shown that θ is a congruence, each θ-class is a complete sublattice of λ(S) and the maximum element in each θ-class is determined. 1980 Mathematics subject classification (Amer. Math. Soc.): 20 M 10.

We determine which permutative varieties are saturated and classify all nontrivial permutation identities for the class of all globally idempotent semigroups.

Completely simple semigroups form a variety, , of algebras with the operations of multiplication and inversion. It is known that the mapping , where is the variety of all groups, is an isomorphism of the lattice of all subvarieties of onto a subdirect product of the lattice of subvarieties of and the interval . We consider embeddings of into certain direct products on the above pattern with rectangular bands, rectangular groups and central completely simple semigroups in place of groups.

A verbal product is introduced for a particular class of varieties of inverse semigroups and this product is shown to be associative. As well, the structure of this class is examined.

A new arrow notation is used to describe biordered sets. Biordered sets are characterized as biordered subsets of the partial algebras formed by the idempotents of semigroups. Thus it can be shown that in the free semigroup on a biordered set factored out by the equations of the biordered set there is no collapse of idempotents and no new arrows.

We find the atoms of certain subclasses of varieties of finite semigroups and the corresponding varieties of languages. For example we give a new description of languages whose syntactic monoids are R-trivial and idempotent. We also describe the least variety containing all commutative semigroups and at least one non-commutative semigroup. Finally we extend to varieties of finite semigroups some classical results about semilattice decomposition of semigroups.

A variant of Kurosh-Amitsur radical theory is developed for algebras with a collection of (finitary) operations ω, all of which are idempotent, that is satisfy the condition ω(x, x,…, x) = x. In such algebras, all classes of any congruence are subalgebras. In place of a largest normal radical subobject, a largest congruence with radical congruence classes is considered. In congruence-permutable varieties the parallels with conventional radical theory are most striking.