A DR-semigroup S (also known as a reduced E-semiabundant or reduced E-Fountain semigroup) is here viewed as a semigroup equipped with two unary operations D, R satisfying finitely many equational laws. Examples include DRC-semigroups (hence Ehresmann semigroups), which also satisfy the congruence conditions. The ample conditions on DR-semigroups are studied here and are defined by the laws xD(y)=D(xD(y))x and R(y)x=xR(R(y)x). Two natural partial orders may be defined on a DR-semigroup and we show that the ample conditions hold if and only if the two orders are equal and the projections (elements of the form D(x)) commute with one-another. Restriction semigroups satisfy the ample conditions, but we give non-restriction examples using closure operators on sets, strongly order-preserving functions on a quasiordered set, and certain subsets of partial categories. Following the work of Stein, we show how to construct a certain partial algebra C(S) from any DR-semigroup, which is a category if S satisfies the congruence conditions, but is “almost" a category if the ample conditions hold. We then characterise the ample conditions in terms of the equality of two natural partial orders, and in terms of a converse of the condition on S ensuring that C(S) is a category. Our main result is an ESN-style theorem for DR-semigroups satisfying the ample conditions, based on the C(S) construction. We also obtain an embedding theorem, generalizing a result for restriction semigroups due to Lawson.

We construct the lower radical class and the semisimple closure for a given class using class operators and detail some of the properties of these operators and their interplay with the operators already used in radical theory. The setting is the class of algebras introduced by Puczy lowski which ensures the results hold in groups, multi-operator groups such as rings, as well as loops and hoops.

We give an algebraic characterisation of ordered groupoids, namely, we show that there is a categorical isomophism between the category of ordered groupoids and the category of $D$-inverse constellations. Here constellations are partial algebras in the sense that they possess a partial product, and a unary operation $D$. We consider constellations in which elements have a suitable notion of inverse, giving the notion of a D-inverse constellation.

The override operation ⊔ is a natural one in computer science, and has connections with other areas of mathematics such as hyperplane arrangements. For arbitrary functions f and g, f⊔ g is the function with domain dom (f)∪ dom (g) that agrees with f on dom (f) and with g on dom (g) \ dom (f) . Jackson and the author have shown that there is no finite axiomatisation of algebras of functions of signature (⊔ ) . But adding operations (such as update) to this minimal signature can lead to finite axiomatisations. For the functional signature (⊔ ,\ ) where \ is set-theoretic difference, Cirulis has given a finite equational axiomatisation as subtraction o-semilattices. Define f⋎ g=(f⊔ g)∩ (g⊔ f) for all functions f and g; this is the largest domain restriction of the binary relation f∪ g that gives a partial function. Now f∩ g=f\ (f\ g) and f⊔ g=f⋎ (f⋎ g) for all functions f, g, so the signatures (⋎ ) and (⊔ ,∩ ) are both intermediate between (⊔ ) and (⊔ ,\ ) in expressive power. We show that each is finitely axiomatised, with the former giving a proper quasivariety and the latter the variety of associative distributive o-semilattices in the sense of Cirulis.

We obtain an ESN theorem for a very general class of biunary semigroups with idempotent-valued domain and range operations, representing them in terms of small categories equipped with a suitable biaction of the identities on the category. Our results generalise the recent work of Fitzgerald and Kinyon connecting localisable semigroups to transcription categories, as well as that of Lawson linking Ehresmann semigroups to categories with Ehresmann biaction. In contrast to most approaches to ESN theorems, we do not require the categories to be ordered or for their sets of identities to possess any particular structure. Throughout, the biunary semigroups are represented using categories rather than generalised categories of any kind, and we obtain category isomorphisms between the clesses of semigroups and their associated enriched categories, rather than category equivalences. Our results cover the class of DRC-semigroups considered by Jones and Shoufeng Wang, but they also cover cases where not both congruence conditions hold, including examples such as the semigroup of binary relations on a set under demonic composition equipped with domain and range operations.

Motivated by interior operators and by analogous generalisations of closure operators to semigroups in general, we define left, right and two-sided interior operations on a semigroup S in terms of a distinguished set of idempotents E using the natural order. In the left-sided case, we require that for all s∈ S , there is a largest e∈ E under the natural order such that es=e . We generalise them using suitable Green-like relations, and characterise the resulting classes of unary and biunary semigroups as quasivarieties or varieties.

We provide complete classifications of algebras of partial maps for a significant swathe of combinations of operations not previously classified. Our focus is the many subsidiary operations that arise in recent considerations of the ‘override’ and ‘update’ operations arising in specification languages. These other operations turn out to have an older pedigree: domain restriction, set subtraction and intersection. All signatures considered include domain restriction, at least as a term. Combinations of the operations are classified and given complete axiomatizations with and without the presence of functional composition. Each classification is achieved by way of providing a concrete representation of the corresponding abstract algebras as partial maps acting on special kinds of filters determined with respect to various induced orders. In contrast to many negative results in the broader area, all of the considered combinations lead to finite axiomatizations.

Given a monoid S with E any non-empty subset of its idempotents, we present a novel one-sided version of idempotent completion we call left E-completion. In general, the construction yields a one-sided variant of a small category called a constellation by Gould and Hollings. Under certain conditions, this constellation is inductive, meaning that its partial multiplication may be extended to give a left restriction semigroup, a type of unary semigroup whose unary operation models domain. We study the properties of those pairs S,E for which this happens, and characterise those left restriction semigroups that arise as such left E-completions of their submonoid of elements having domain 1. As first applications, we decompose the left restriction semigroup of partial functions on the set X and the right restriction semigroup of left total partitions on X as left and right E-completions respectively of the transformation semigroup TX on X, and decompose the left restriction semigroup of binary relations on X under demonic composition as a left E-completion of the left-total binary relations. In many cases, including these three examples, the construction embeds in a semigroup Zappa-Szép product.

On a semigroup S, define the equivalence relation $$\begin{aligned} {{\mathscr {F}}}=\{(a,b)\in S\times S\mid \forall x\in S: xa=x \Leftrightarrow xb=x\}, \end{aligned}$$ and define $${{\mathscr {G}}}$$ dually. We say S is $${{\mathscr {F}}}$$ -abundant if there is an idempotent in every $${{\mathscr {F}}}$$ -class, and similarly for $${{\mathscr {G}}}$$ -abundance, and we say S is $$({{\mathscr {F}}},{{\mathscr {G}}})$$ -abundant if it is both $${{\mathscr {F}}}$$ -abundant and $${{\mathscr {G}}}$$ -abundant. These concepts are analogous to the notions of regularity and one- and two-sided abundance, defined in terms of Green’s relations $${{\mathscr {L}}}$$ and $${{\mathscr {R}}}$$ , and their generalisations $${{\mathscr {L}}}^*$$ and $${{\mathscr {R}}}^*$$ , respectively. We relate this new form of abundance to the earlier ones, considering in particular the analogs of superabundance and amiability.

Ehresmann semigroups may be viewed as biunary semigroups equipped with domain and range operations satisfying some equational laws. Motivated by some of the main examples, we here define ordered Ehresmann semigroups, and consider their basic properties as well as special cases in which the order is algebraically definable. In particular, one and two-sided restriction semigroups equipped with their natural orders are characterised within the class of ordered Ehresmann semigroups. The main result is an ESN-style theorem for ordered Ehresmann semigroups with particular reference to the special cases.

Demonic composition, demonic refinement and demonic union are alternatives to the usual "angelic" composition, angelic refinement (inclusion) and angelic (usual) union defined on binary relations. We first motivate both the angelic and demonic via an analysis of the behaviour of non-deterministic programs, with the angelic associated with partial correctness and demonic with total correctness, both cases emerging from a richer algebraic model of non-deterministic programs incorporating both aspects. Zareckii has shown that the isomorphism class of algebras of binary relations under angelic composition and inclusion is finitely axiomatised as the class of ordered semigroups. The proof can be used to establish that the same axiomatisation applies to binary relations under demonic composition and refinement, and a further modification of the proof can be used to incorporate a zero element representing the empty relation in the angelic case and the full relation in the demonic case. For the signature of angelic composition and union, it is known that no finite axiomatisation exists, and we show the analogous result for demonic composition and demonic union by showing that the same axiomatisation holds for both. We show that the isomorphism class of algebras of binary relations with the "mixed" signature of demonic composition and angelic inclusion has no finite axiomatisation. As a contrast, we show that the isomorphism class of partial algebras of binary relations with the partial operation of constellation product and inclusion (also a "mixed" signature) is finitely axiomatisable.

On a semigroup S , define the equivalence relation ℱ={(a,b)∈ S× S|∀ x∈ S: xa=x ⇔ xb=x}, and define 𝒢 dually. We say S is ℱ - abundant if there is an idempotent in every ℱ -class, and similarly for 𝒢 -abundance, and we say S is (ℱ,𝒢) -abundant if it is both ℱ -abundant and 𝒢 -abundant. These concepts are analogous to the notions of regularity and one- and two-sided abundance, defined in terms of Green’s relations ℒ and ℛ , and their generalisations ℒ^* and ℛ^* , respectively. We relate this new form of abundance to the earlier ones, considering in particular the analogs of superabundance and amiability.

Constellations are asymmetric generalisations of categories. Although they are not required to possess a notion of range, many natural examples do. These include commonly occurring constellations related to concrete categories (since they model surjective morphisms), and also others arising from quite different sources, including from well-studied classes of semigroups. We show how constellations with a well-behaved range operation are nothing but ordered categories with restrictions. We characterise abstractly those categories that are canonical extensions of constellations with range, as so-called IS-categories. Such categories contain distinguished subcategories of insertions (which are monomorphisms) and surjections (in general different to the epimorphisms) such that each morphism admits a unique factorisation into a surjection followed by an insertion. Most familiar concrete categories are IS-categories, and we show how some of the well-known properties of these categories arise from the fact that they are IS-categories. For appropriate choices of morphisms in each, the category of IS-categories is shown to be equivalent to the category of constellations with range.

Override and update are natural constructions for combining partial functions, which arise in various program specification contexts. We use an unexpected connection with combinatorial geometry to provide a complete finite system of equational axioms for the first order theory of the override and update constructions on partial functions, resolving the main unsolved problem in the area.

Demonic composition $$*$$ is an associative operation on binary relations, and demonic refinement $${\sqsubseteq }$$ is a partial order on binary relations. Other operations on binary relations considered here include the unary domain operation D and the left restrictive multiplication operation $$\circ $$ given by $$s\circ t=D(s)*t$$ . We show that the class of relation algebras of signature $$\{\, \sqsubseteq , D, *\, \}$$ , or equivalently $$\{\, \subseteq , \circ , *\, \}$$ , has no finite axiomatisation. A large number of other non-finite axiomatisability consequences of this result are also given, along with some further negative results for related signatures. On the positive side, a finite set of axioms is obtained for relation algebras with signature $$\{\, \sqsubseteq , \circ , *\, \}$$ , hence also for $$\{\, \subseteq , \circ , *\, \}$$ .

Demonic composition is defined on the set of binary relations over the non-empty set X, $$Rel_X$$ , and is a variant of standard or “angelic” composition. It arises naturally in the setting of the theory of non-deterministic computer programs, and shares many of the nice features of ordinary composition (it is associative, and generalises composition of functions). When equipped with the operations of demonic composition and domain, $$Rel_X$$ is a left restriction semigroup (like $$PT_X$$ , the semigroup of partial functions on X), whereas usual composition and domain give a unary semigroup satisfying weaker laws. By viewing $$Rel_X$$ under a restricted version of its usual composition and domain as a constellation (a kind of “one-sided” category), we show how this demonic left restriction semigroup structure arises on $$Rel_X$$ , placing it in a more general context. The construction applies to any unary semigroup with a “domain-like” operation satisfying certain minimal conditions which we identify. In particular it is shown that using the construction, any Baer $$*$$ -semigroup S can be given a left restriction semigroup structure which is even an inverse semigroup if S is $$*$$ -regular. It follows that the semigroup of $$n\times n$$ matrices over the real or complex numbers is an inverse semigroup with respect to a modified notion of product that almost always agrees with the usual matrix product, and in which inverse is pseudoinverse (Moore–Penrose inverse).

This paper establishes a finite axiomatization of possibly non-halting computer programs and tests, with the if-then-else operation. The model is a two-sorted algebra, with one sort being the programs and the other being the tests. The main operation on programs is composition, and 1 and 0 represent the programs skip and loop (i.e. never halts) respectively. Programs are modeled as partial functions on some state space [Formula: see text], with tests modeled as partial predicates on [Formula: see text]. The operations on the tests are the usual logical connectives ∧, ∨, [Formula: see text], [Formula: see text] and [Formula: see text]. In addition, there is the hybrid operation of if-then-else, and the test-valued operation [Formula: see text] on programs which is true when a program halts, and undefined otherwise. The halting operation [Formula: see text] implies that operations of domain [Formula: see text] and domain join ∨ may also be expressed. When tests are assumed to be possibly non-halting, the evaluation strategy of the logical connectives affects the result. Here we model parallel evaluation, as opposed to the common sequential (or short-circuit) evaluation strategy. For example, we view [Formula: see text] as false if either [Formula: see text] or [Formula: see text] is false, even if the other does not halt.

A generalised D-semigroup is here defined to be a left E-semiabundant semigroup S in which the \(\overline{\mathcal R}_E\)-class of every \(x\in S\) contains a unique element D(x) of E, made into a unary semigroup. Two-sided versions are defined in the obvious way in terms of \(\overline{\mathcal R}_E\) and \(\overline{\mathcal L}_E\). The resulting class of unary (bi-unary) semigroups is shown to be a finitely based variety, properly containing the variety of D-semigroups (defined in an order-theoretic way in Communications in Algebra, 3979–4007, 2014). Important subclasses associated with the regularity and abundance properties are considered. The full transformation semigroup \(T_X\) can be made into a generalised D-semigroup in many natural ways, and an embedding theorem is given. A generalisation of inverse semigroups in which inverses are defined relative to a set of idempotents arises as a special case, and a finite equational axiomatisation of the resulting unary semigroups is given.

In a result generalising the Ehresmann–Schein–Nambooripad Theorem relating inverse semigroups to inductive groupoids, Lawson has shown that Ehresmann semigroups correspond to certain types of ordered (small) categories he calls Ehresmann categories. An important special case of this is the correspondence between two-sided restriction semigroups and what Lawson calls inductive categories. Gould and Hollings obtained a one-sided version of this last result, by establishing a similar correspondence between left restriction semigroups and certain ordered partial algebras they call inductive constellations (a general constellation is a one-sided generalisation of a category). We put this one-sided correspondence into a rather broader setting, at its most general involving left congruence D-semigroups (which need not satisfy any semiadequacy condition) and what we call co-restriction constellations, a finitely axiomatized class of partial algebras. There are ordered and unordered versions of our results. Two special cases have particular interest. One is that the class of left Ehresmann semigroups (the natural one-sided versions of Lawson’s Ehresmann semigroups) corresponds to the class of co-restriction constellations satisfying a suitable semiadequacy condition. The other is that the class of ordered left Ehresmann semigroups (which generalise left restriction semigroups and for which semigroups of binary relations equipped with domain operation and the inclusion order are important examples) corresponds to a class of ordered constellations defined by a straightforward weakening of the inductive constellation axioms.

Constellations are partial algebras that are one-sided generalisations of categories. It has previously been shown that the category of inductive constellations is isomorphic to the category of left restriction semigroups. Here we consider constellations in full generality, giving many examples. We characterise those small constellations that are isomorphic to constellations of partial functions. We examine in detail the relationship between constellations and categories, showing the latter to be special cases of the former. In particular, we characterise those constellations that arise as (sub-)reducts of categories, and show that categories are nothing but two-sided constellations. We demonstrate that the notion of substructure can be captured within constellations but not within categories. We show that every constellation $P$ gives rise to a category $\mathcal{C}(P)$, its canonical extension, in a simplest possible way, and that $P$ is a quotient of $\mathcal{C}(P)$ in a natural sense. We also show that many of the most common concrete categories may be constructed from simpler quotient constellations using this construction. We characterise the canonical congruences $\delta$ on a given category $K$ (those for which $K\cong \mathcal{C}((K/\delta)$), and show that the category of constellations is equivalent to the category of categories equipped with distinguished canonical congruence.