On lattices of varieties of restriction semigroups

Abstract

The left restriction semigroups have arisen in a number of contexts, one being as the abstract characterization of semigroups of partial maps, another as the ‘weakly left E-ample’ semigroups of the ‘York school’, and, more recently as a variety of unary semigroups defined by a set of simple identities. We initiate a study of the lattice of varieties of such semigroups and, in parallel, of their two-sided versions, the restriction semigroups. Although at the very bottom of the respective lattices the behaviour is akin to that of varieties of inverse semigroups, more interesting features are soon found in the minimal varieties that do not consist of semilattices of monoids, associated with certain ‘forbidden’ semigroups. There are two such in the one-sided case, three in the two-sided case. Also of interest in the one-sided case are the varieties consisting of unions of monoids, far indeed from any analogue for inverse semigroups. In a sequel, the author will show, in the two-sided case, that some rather surprising behavior is observed at the next ‘level’ of the lattice of varieties.

Appendix

The material in this paper is self-contained, in that only the defining identities are needed. Gould [11] was the first to make explicit the identification of the varietal definitions of [left] restriction semigroups with the ‘traditional’ definitions of weakly [left] E-ample semigroups, and it was her paper that motivated the author to investigate the lattices of varieties. The later paper by Hollings [13] surveyed ‘the historical development of the study of left restriction semigroups, from the ‘weakly left E-ample’ perspective’, taking as the definition of left restriction semigroups, however, the semigroups of partial mappings of a given set that are closed under taking the identity maps on their domains.

Together, those two papers demonstrate the equivalence of these three approaches to the topic. They also provide a broad overview of the development of the various historical strands of development of the topic, including some not touched upon here, to which we refer the reader.

Here we briefly summarize these equivalences, so as to place our paper in context. Naturally, the reader is referred to [11] and [13] for a fuller exposition.

Let S be a semigroup and let E be a nonempty ‘distinguished’ subsemilattice of E _S . Define the relation $\widetilde {\mathrel {\mathcal {R}}}_{E}$ on S by $a \mathrel{\widetilde {\mathrel {\mathcal {R}}}_{E}} b$ if, for all e∈E, ea=a if and only if eb=b. Each $\widetilde {\mathrel {\mathcal {R}}}_{E}$-class of S contains at most one member of E. Call S weakly left E-ample if

• (1) every element a of S is $\widetilde {\mathrel {\mathcal {R}}}_{E}$-related to a (necessarily unique) member of E, which may be denoted a ⁺;

• (2) $\widetilde {\mathrel {\mathcal {R}}}_{E}$ is a left congruence;

• (3) for all a∈S, e∈E, ae=(ae)⁺ a.

Treating S now as a unary semigroup (S,⋅,⁺), and referring to the defining identities for left restriction semigroups in Sect. 2, notice that E={x∈S:x ⁺=x}, so (x ⁺)⁺=x ⁺ holds, the identities x ⁺ x=x and x ⁺ y ⁺=y ⁺ x ⁺ are obvious, the identity (xy)⁺=(xy ⁺)⁺ (see Lemma 2.1) follows from (2), and the left ample identity xy ⁺=(xy)⁺ x is an immediate consequence. Also as a result of that additional identity, (x ⁺ y)⁺=(x ⁺ y ⁺)⁺=x ⁺ y ⁺.

Therefore every weakly left E-ample semigroup, regarded as a unary semigroup, satisfies the identities that we have used to define left restriction semigroups, and E is its set of projections. Conversely, given any left restriction semigroup (S,⋅,⁺) and putting E=P _S , then $a \mathrel{\widetilde {\mathrel {\mathcal {R}}}_{E}} b$ if and only if a ⁺=b ⁺, that is, aℝb in our notation, from which it readily follows that S is weakly left E-ample.

When regarded from the varietal point of view, the semilattice of ‘distinguished idempotents’ is now no longer ‘distinguished’: it is simply the semilattice of projections, subsidiary to the unary operation. Thus the subscript notation on the generalized Green’s relations plays only the historical role of distinguishing these semigroups from the earlier classes considered in the next paragraph. That is why we have chosen to start afresh with the notation ℝ, etc. A further reason is that these relations do behave in many ways like the ‘usual’ Green’s relations, as will be more clearly seen in the sequel [17], where ‘partial egg-boxes’ will play a central role.

The term weakly left ample is reserved for the special case that E=E _S . The term left ample refers to the case that $\widetilde {\mathrel {\mathcal {R}}}_{E}= \mathrel {\mathcal {R}}^{*}$, the ‘potential’ Green’s relation given by $a \mathrel {\mathcal {R}}^{*} b$ if xa=ya if and only if xb=yb for all x,y∈S ¹. (Necessarily, E=E _S [11].) The inverse semigroups, and their full subsemigroups, provide a ready source of left ample semigroups. (See also Result 5.1 below.)

From the universal algebraic point of view, the great advantage of working with left restriction semigroups is that they form a variety. The weakly left ample and the left ample semigroups form only quasi-varieties. At least in the author’s view, they also exhibit the most natural generality, in that the reduced left restriction semigroups comprise all monoids, whereas in the case of weakly left ample and left ample semigroups, they yield instead the unipotent and the right cancellative monoids, respectively, (see, for example, [10, Proposition 2.5]).

The explicit correspondence between weakly left E-ample semigroups and semigroups of partial mappings, which goes back in its essence to Trokhimenko [23], may also be found in [11, 13]. Denote by $\mathcal{PT}_{X}$ the semigroup of partial mappings of a nonempty set X, under composition, and for $\alpha \in \mathcal{PT}_{X}$, let α ⁺ be the identity map on the domain of α. Within $\mathcal{PT}_{X}$ lies the inverse semigroup $\mathcal{I}_{X}$ of partial one-one mappings of X, under the natural inverse.

Result 4.23

The unary semigroup $(\mathcal{PT}s_{X}, \circ, ^{+})$ is a weakly left E-ample semigroup, the semilattice of projections consisting of the identity mappings on subsets of X. Conversely, any weakly left E-ample semigroup is (unarily) isomorphic to a unary subsemigroup of such a semigroup. The representation is by one-one mappings if and only if the semigroup is left ample.

The two-sided connections are established similarly. However, there is apparently no two-sided analogue of this last result.