On EQ-monoids

An EQ-monoid A is a monoid with distinguished subsemilattice L with 1 2 L and such that any a, b 2 A have a largest right equalizer in L. The class of all such monoids equipped with a binary operation that identifies this largest right equalizer is a variety. Examples include Heyting algebras, Cartesian products of monoids with zero, as well as monoids of relations and partial maps on sets. The variety is 0-regular (though not ideal determined and hence congruences do not permute), and we describe the normal subobjects in terms of a global semilattice structure. We give representation theorems for several natural subvarieties in terms of Boolean algebras, Cartesian products and partial maps. The case in which the EQmonoid is assumed to be an inverse semigroup with zero is given particular attention. Finally, we define the derived category associated with a monoid having a distinguished subsemilattice containing the identity (a construction generalising the idea of a monoid category), and show that those monoids for which this derived category has equalizers in the semilattice constitute a variety of EQ-monoids

Semigroups with if-then-else and halting programs

The "if–then–else" construction is one of the most elementary programming commands, and its abstract laws have been widely studied, starting with McCarthy. Possibly, the most obvious extension of this is to include the operation of composition of programs, which gives a semigroup of functions (total, partial, or possibly general binary relations) that can be recombined using if–then–else. We show that this particular extension admits no finite complete axiomatization and instead focus on the case where composition of functions with predicates is also allowed (and we argue there is good reason to take this approach). In the case of total functions — modeling halting programs — we give a complete axiomatization for the theory in terms of a finite system of equations. We obtain a similar result when an operation of equality test and/or fixed point test is included

Identities in the Algebra of Partial Maps

We consider the identities of a variety of semigroup-related algebras modelling the algebra of partial maps. We show that the identities are intimately related to a weak semigroup deductive system and we show that the equational theory is decidable. We do this by giving a term rewriting system for the variety. We then show that this variety has many subvarieties whose equational theory interprets the full uniform word problem for semigroups and consequently are undecidable. As a corollary it is shown that the equational theory of Clifford semigroups whose natural order is a semilattice is undecidable

Monoids with tests and the algebra of possibly non-halting programs

We study the algebraic theory of computable functions, which can be viewed as arising from possibly non-halting computer programs or algorithms, acting on some state space, equipped with operations of composition, if-then-else and while-do defined in terms of a Boolean algebra of conditions. It has previously been shown that there is no finite axiomatisation of algebras of partial functions under these operations alone, and this holds even if one restricts attention to transformations (representing halting programs) rather than partial functions, and omits while-do from the signature. In the halting case, there is a natural “fix”, which is to allow composition of halting programs with conditions, and then the resulting algebras admit a finite axiomatisation. In the current setting such compositions are not possible, but by extending the notion of if-then-else, we are able to give finite axiomatisations of the resulting algebras of (partial) functions, with while-do in the signature if the state space is assumed finite. The axiomatisations are extended to consider the partial predicate of equality. All algebras considered turn out to be enrichments of the notion of a (one-sided) restriction semigrou

Partial maps with domain and range: extending Schein's representation

The semigroup of all partial maps on a set under the operation of composition admits a number of operations relating to the domain and range of a partial map. Of particular interest are the operations R and L returning the identity on the domain of a map and on the range of a map respectively. Schein [25] gave an axiomatic characterisation of the semigroups with R and L representable as systems of partial maps; the class is a finitely axiomatisable quasivariety closely related to ample semigroups (which were introduced—as type A semigroups—by Fountain, [7]). We provide an account of Schein's result (which until now appears only in Russian) and extend Schein's method to include the binary operations of intersection, of greatest common range restriction, and some unary operations relating to the set of fixed points of a partial map. Unlike the case of semigroups with R and L, a number of the possibilities can be equationally axiomatised

Radicals of 0-regular algebras

We consider a generalisation of the Kurosh--Amitsur radical theory for rings (and more generally multi-operator groups) which applies to 0-regular varieties in which all operations preserve 0. We obtain results for subvarieties, quasivarieties and element-wise equationally defined classes. A number of examples of radical and semisimple classes in particular varieties are given, including hoops, loops and similar structures. In the first section, we introduce 0-normal varieties (0-regular varieties in which all operations preserve 0), and show that a key isomorphism theorem holds in a 0-normal variety if it is subtractive, a property more general than congruence permutability. We then define our notion of a radical class in the second section. A number of basic results and characterisations of radical and semisimple classes are then obtained, largely based on the more general categorical framework of L. M\'arki, R. Mlitz and R. Wiegandt as in [13]. We consider the subtractive case separately. In the third section, we obtain results concerning subvarieties and quasivarieties based on the results of the previous section, and also generalise to subtractive varieties some results for multi-operator group radicals defined by simple equational rules. Several examples of radical and semisimple classes are given for a range of fairly natural 0-normal varieties of algebras, most of which are subtractive

Constellations and their relationship with categories

Constellations are partial algebras that are one-sided generalisations of categories. Indeed, we show that a category is exactly a constellation that also satisfies the left-right dual axioms. Constellations have previously appeared in the context of inductive constellations: the category of inductive constellations is known to be 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. In particular, we characterise those constellations that arise as (sub-)reducts of categories. 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 (Formula presented.), its canonical extension, in a simplest possible way, and that P is a quotient of (Formula presented.) 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 (Formula presented.) on a given category (Formula presented.) (those for which (Formula presented.), and show that the category of constellations is equivalent to the category of (Formula presented.)-categories, that is, categories equipped with distinguished canonical congruence (Formula presented.). The main observation of this paper is that category theory as it applies to the familiar concrete categories of modern mathematics (which come equipped with natural notions of substructures and indeed are (Formula presented.)-categories) may be subsumed by constellation theory

An Overview of the Karst Areas in British Columbia, Canada

Karst is a three-dimensional landscape that occurs in soluble bedrock (typically limestone, marble, dolostone, gypsum or halite) and is defined by a solutionally weathered surface, a subsurface drainage system (where conduit-flow dominates), and underground openings and caves. Karst can host unique flora and subsurface fauna, as well as a wide range of other scientific, recreational and cultural values. Karst and potential karst areas underlie approximately 10% of British Columbia (BC), but the distribution and extent of this landscape has yet to be fully explored and delineated. Some of the most extensive and well-developed karst areas occur within the forestedlimestone areas of coastal BC, such as on Vancouver Island and Haida Gwaii, where numerous surface karst features and caves are known. Karst in the interior plateau regions of British Columbia is less well known, being in part covered by thick deposits of glacial materials. Alpine karst regions are most apparent in the Rocky Mountains where there are limestone plateaus, karst drainages and cave systems that have close connections to past and present glacial systems. Mapping of karst is a critical component for any land-use or resource development activity in all regions of British Columbia, as the environmental impacts on karst and its associated values are potentially significant. The regional distribution of karst in BC is not well mapped, with only an office-based reconnaissance karst potential map (1:250,000-scale) and a related database completed in 1999. A renewed effort should now be made to better map karst across British Columbia using digital bedrock mapping data released in 2017, combined with more recent satellite imagery and improved field knowledge.RÉSUMÉLe karst est un paysage tridimensionnel qui se présente dans le substrat rocheux soluble (généralement calcaire, marbre, dolomite, gypse ou halite) et est défini par une surface altérée par dissolution, un système de drainage souterrain (où l’écoulement par conduit domine) et des ouvertures et cavernes souterraines. Le karst peut abriter une flore et une faune souterraine unique, ainsi qu’une grande variété d’autres ressources scientifiques, de loisir et culturelles. Les zones karstiques et potentiellement karstiques constituent environ 10% de la surface de la Colombie-Britannique, mais la répartition et l’étendue de ce paysage n’a pas été complètement explorées et circonscrites. Certaines des zones karstiques les plus étendues et les mieux développées se trouvent dans les calcaires des zones calcaires boisées de la côte de la Colombie-Britannique, telles que l’île de Vancouver et l’archipel de Haida Gwaii, où l’on connaît de nombreuses caractéristiques karstiques de surface et des cavernes. Le karst des régions des plateaux intérieurs de la Colombie-Britannique est moins bien connu, étant en partie recouvert d’épais dépôts de matériaux glaciaires. Les régions karstiques alpines sont plus apparentes dans les montagnes Rocheuses où se trouvent des plateaux calcaires, des bassins de drainage karstiques et des systèmes de cavernes étroitement liés aux systèmes glaciaires passés et contemporains. La cartographie du karst est une constituante essentielle de toute activité d’utilisation du terrain ou de développement des ressources dans toutes les régions de la Colombie-Britannique, car les impacts environnementaux sur le karst et ses bénéfices associés sont potentiellement importants. La distribution régionale et les caractéristiques des karsts en Colombie-Britannique ne sont pas bien cartographiées, avec seulement une carte de reconnaissance du potentiel karstique établie par une étude de bureau (à l’échelle de 1/250 000) et une base de données associée, complétées en 1999. Il faut aujourd’hui améliorer la cartographie de karsts en Colombie-Britannique en utilisant les données numériques de cartographie du substrat rocheux publiées en 2017, combinées avec des images satellite plus récentes et à une meilleure connaissance du terrain