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This paper is a direct continuation of the paper “Duality for dyadic intervals” by the same authors, and can be considered as its second part. Dyadic rationals are rationals whose denominator is a power of 2. Dyadic triangles and dyadic polygons are, respectively, defined as the intersections with the dyadic plane of a triangle or polygon in the real plane whose vertices lie in the dyadic plane. The one-dimensional analogues are dyadic intervals. Algebraically, dyadic polygons carry the structure of a commutative, entropic and idempotent groupoid under the binary operation of arithmetic mean. The first paper dealt with the structure of finitely generated subgroupoids of the dyadic line, which were shown to be isomorphic to dyadic intervals. Then a duality between the class of dyadic intervals and the class of certain subgroupoids of the dyadic unit square was described. The present paper extends the results of the first paper, provides some characterizations of dyadic triangles, and describes a duality for the class of dyadic triangles. As in the case of intervals, the duality is given by an infinite dualizing (schizophrenic) object, the dyadic unit interval. The dual spaces are certain subgroupoids of the dyadic unit cube, considered as (commutative, idempotent and entropic) groupoids with additional constant operations.

The Mal’tsev product of two varieties of similar algebras is always a quasivariety. We consider the question of when this quasivariety is a variety. The main result asserts that if $$\mathcal {V}$$ is a strongly irregular variety with no nullary operations and at least one non-unary operation, and $$\mathcal {S}$$ is the variety, of the same type as $$\mathcal {V}$$ , equivalent to the variety of semilattices, then the Mal’tsev product $$\mathcal {V}\circ \mathcal {S}$$ is a variety. It consists precisely of semilattice sums of algebras in $$\mathcal {V}$$ . We derive an equational base for the product from an equational base for $$\mathcal {V}$$ . However, if $$\mathcal {V}$$ is a regular variety, then the Mal’tsev product may not be a variety. We discuss various applications of the main result, and examine some detailed representations of algebras in $$\mathcal {V}\circ \mathcal {S}$$ .

Barycentric algebras are fundamental for modeling convex sets, semilattices, affine spaces and related structures. This paper is part of a series examining the concept of a barycentric algebra in detail. In preceding work, threshold barycentric algebras were introduced as part of an analysis of the axiomatization of convexity. In the current paper, the concept of a threshold barycentric algebra is extended to threshold affine spaces. To within equivalence, these algebras comprise barycentric algebras, commutative idempotent entropic magmas, and affine spaces, all defined over a subfield of the field of real numbers. Many properties of threshold barycentric algebras extend to threshold affine spaces.

Wille’s formal concept analysis, Hardegree’s treatment of natural kinds, and Birkhoff’s mathematical theory of polarities provide essentially equivalent tools for the analysis of a static system functioning at a single level. We now show how a number of categorical notions allow these tools to be extended to cover the analysis of complex systems involving multiple hierarchical levels indexed by a semilattice, including the case where a chain represents a time series governing the evolution of a single system. A semilattice is a poset category with finite products or coproducts. Our analysis then rests on functors from a semilattice to the category of complete lattice homomorphisms, or from a semilattice to a category of polarities and bonding relations.

In an earlier paper, Romanowska, Ślusarski and Smith described a duality between the category of polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations. The present paper provides an extension of this duality to a much more general class of so-called quasipolytopes, that is, convex sets with polytopes as closures. The dual spaces of quasipolytopes are Płonka sums of open polytopes, which are considered as barycentric algebras with some additional operations. In constructing this duality, we use several known and new dualities: the Hofmann–Mislove–Stralka duality for semilattices; the Romanowska–Ślusarski–Smith duality for polytopes; a new duality for open polytopes; and a new duality for injective Płonka sums of polytopes.

Convex sets may be viewed as algebras equipped with a set of binary convex combinations that is indexed by the open unit interval of real numbers. Convex sets generate the variety of barycentric algebras, which also includes semilattices where the semilattice multiplication is repeated uncountably many times. Barycentric algebras are defined by three axioms: idempotence, skew-commutativity, and skew-associativity. Since the skew-associativity axiom is rather complicated, Klaus Keimel has asked whether it can simply be replaced by the entropic law. It turns out that the answer is negative. The counterexamples presented and studied in this paper are known as threshold barycentric algebras, depending on a threshold taken from the left-hand side of the closed unit interval. They offer an entire spectrum of algebras, ranging from barycentric algebras for threshold 0 to commutative idempotent entropic groupoids for threshold 1/2.

Comonoid, bi-algebra, and Hopf algebra structures are stud-ied within the universal-algebraic context of entropic varieties. Attentionfocuses on the behavior of setlike and primitive elements. It is shown thatentropic Jo´nsson-Tarski varieties provide a natural universal-algebraicsetting for primitive elements and group quantum couples (generaliza-tions of the group quantum double). Here, the set of primitive elementsof a Hopf algebra forms a Lie algebra, and the tensor algebra on anyalgebra is a bi-algebra. If the tensor algebra is a Hopf algebra, then theunderlying Jo´nsson-Tarski monoid of the generating algebra is cancella-tive. The problem of determining when the Jo´nsson-Tarski monoid formsa group is open. 1. IntroductionThe aim of this paper is to consider comonoid, bi-algebra, and Hopf alge-bra structure within the context of universal algebra. The general categoricalsetting for such structure is provided by symmetric monoidal categories (or“symmetric tensor categories”, compare [11, p. 69]), with an associative tensorproduct and a unit object. Now an algebra (A,Ω) of type τ : Ω → Nis said tobe entropic if the operationω: A

Modes are idempotent and entropic algebras. While the mode structure of sets of submodes has received considerable attention in the past, this paper is devoted to the study of mode structure on sets of mode homomorphisms. Connections between the two constructions are established. A detailed analysis is given for the algebra of homomorphisms from submodes of one mode to submodes of another. In particular, it is shown that such algebras can be decomposed as Płonka sums of more elementary homomorphism algebras. Some critical examples are examined.

Convex subsets of affine spaces over the field of real numbers are described by so-called barycentric algebras. In this paper, we discuss extensions of the geometric and algebraic definitions of a convex set to the case of more general coefficient rings. In particular, we show that the principal ideal subdomains of the reals provide a good framework for such a generalization. Since the closed intervals of these subdomains play an essential role, we provide a detailed analysis of certain cases, and discuss differences from the "classical" intervals of the reals. We introduce a new concept of an algebraic closure of "geometric" convex subsets of affine spaces over the subdomains in question, and investigate their properties. We show that this closure provides a purely algebraic description of topological closures of geometric generalized convex sets. Our closure corresponds to one instance of the very general closure introduced in an earlier paper of the authors. The approach used in this paper allows to extend some results from that paper. Moreover, it provides a very simple description of the closure, with concise proofs of existence and uniqueness.

This paper specifies a detailed, fully type-based general method for translating the class of all pure, many-sorted algebras of a given constant-free type into an equivalent variety of single-sorted algebras of defined, constant-free type. The complexity of the identities defining the variety is a linear function of the number of sorts and the arity of the fundamental operations.

Let A be an algebra (of an arbitrary finitary type), and let γ be a binary term. A pair (a, b) of elements of A will be called a γ-eligible pair if for each x in the subalgebra generated by {a, b} such that x is distinct from a there exists an element y in A such that b = xyγ. We say that A is a γ-closed algebra if for each γ-eligible pair (a, b) there is an element c with b = acγ. We call A a closed algebra if it is γ-closed for all binary terms γ that do not induce a projection.

The problem of embedding modes as subreducts into semimodules over commutative semirings raises the question of characterizing those semimodules that may contain modes as subreducts. We initiate a study of such semimodules by providing certain classes of semimodules with this property. First we characterize mode reducts of commutative monoids, showing that they are equivalent to certain ternary algebras coming from the regularizations of varieties of integral affine spaces, and to some n-ary semilattices. Then we generalize these results, characterizing mode reducts of semimodules over commutative unital rings and semirings obtained from them by adding a new zero element.

Differential modes provide examples of modes that do not embed as subreducts into semimodules over commutative semirings. The current paper studies differential modes, so-called Szendrei differential modes, which actually do embed into semimodules. These algebras form a variety. The main result states that the lattice of nontrivial subvarieties is dually isomorphic to the (nonmodular) lattice of congruences of the free commutative monoid on two generators. Consequently, all varieties of Szendrei differential modes are finitely based.

This paper establishes a duality between the category of polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations.

The paper is devoted to an algebraic and geometric study of the feasible set of a poset, the set of finite probability distributions on the elements of the poset whose weights satisfy the order relationships specified by the poset. For a general poset, this feasible set is a barycentric algebra. The feasible sets of the order structures on a given finite set are precisely the convex unions of the primary simplices, the facets of the first barycentric subdivision of the simplex spanned by the elements of the set. As another fragment of a potential complete duality theory for barycentric algebras, a duality is established between order-preserving mappings and embeddings of feasible sets. In particular, the primary simplices constituting the feasible set of a given finite poset are the feasible sets of the linear extensions of the poset. A finite poset is connected if and only if its barycentre is an extreme point of its feasible set. The feasible set of a (general) disconnected poset is the join of the feasible sets of its components. The extreme points of the feasible set of a finite poset are specified in terms of the disjointly irreducible elements of the semilattice presented by the poset. Semilattices presented by posets are characterised in terms of various distributivity concepts.

A mode (idempotent and entropic algebra) is a Lallement sum of its cancellative submodes over a normal band if it has a congruence with a normal band quotient and cancellative congruence classes. We show that such a sum embeds as a subreduct into a semimodule over a certain ring, and discuss some consequences of this fact. The result generalizes a similar earlier result of the authors proved in the case when the normal band is a semilattice.

In an earlier paper the authors described the isomorphic lattices of quasivarieties of commutative quasigroup modes and of cancellative commutative binary modes. This paper extends the earlier result, and provides a description of the lattice of some irregular quasivarieties of commutative binary modes, i.e. quasivarieties that do not contain semilattices.

The paper describes the isomorphic lattices of quasivarieties of commutative quasigroup modes and of cancellative commutative binary modes. Each quasivariety is characterised by providing a quasi-equational basis. A structural description is also given. Both lattices are uncountable and distributive.

The paper shows that certain sums of algebras admit a representation as subdirect products. As a corollary, one obtains a characterization of subdirectly irreducible algebras. Examples of modes show that the characterization gives a good basis for a detailed investigation of the subdirectly irreducible algebras.

The paper presents general machinery for extending a duality between complete, cocomplete categories to a duality between corresponding categories of semilattice representations (i.e. sheaves over Alexandrov spaces). This enables known dualities to be regularized. Among the applications, regularized Lindenbaum-Tarski duality shows that the weak extension of Boolean logic (i.e. the semantics of PASCAL-like programming languages) is the logic for semilattice-indexed systems of sets. Another application enlarges Pontryagin duality by regularizing it to obtain duality for commutative inverse Clifford monoids.

Snack-algebras - algebras introduced for modelling databases - are shown to be distributive replicas of dissemilattices. (The paper was written while the author visited Iowa State University. Ames. Iowa in Fall 1995.)

The paper discusses “regularisation” of dualities. A given duality between (concrete) categories, e.g. a variety of algebras and a category of representation spaces, is lifted to a duality between the respective categories of semilattice representations in the category of algebras and the category of spaces. In particular, this gives duality for the regularisation of an irregular variety that has a duality. If the type of the variety includes constants, then the regularisation depends critically on the location or absence of constants within the defining identities. The role of schizophrenic objects is discussed, and a number of applications are given. Among these applications are different forms of regularisation of Priestley, Stone and Pontryagin dualities.

and it turns out that the algebra (AS, Q) is again a mode, preserving many of the algebraic properties of (A, ~) [12, 146], [13], [14]. One of the more important examples of modes is given by affine spaces (or affine modules) over ring R ([1], [2], [11], [12], [18]-[211). Affine spaces can be described as modes (E, _R, P) with binary operations _r for each r in R and one ternary Mal'cev (parallelogram-completion) operation P satisfying certain identities

In the first part of this paper, we considered the problem of constructing a (commutative unital) semiring defining the variety of semimodules whose idempotent subreducts lie in a given variety of modes. We provided a general construction of such semirings, along with basic examples and some general properties. In the second part of the paper we discussed some selected varieties of modes, in particular, varieties of affine spaces, varieties of barycentric algebras and varieties of semilattice modes, and described the semirings determining their semi-linearizations, the varieties of semimodules having these algebras as idempotent subreducts. The third part is devoted to varieties of differential groupoids and more general differential modes, and provides the semirings of the semi-linearizations of these varieties.

In the first part of this paper, we considered the prob- lem of constructing a (commutative unital) semiring defining the variety of semimodules whose idempotent subreducts lie in a given variety of modes. We provided a general construction of such semirings, along with basic examples and some general proper- ties. In the second part of the paper we discussed some selected varieties of modes, in particular, varieties of affine spaces, vari- eties of barycentric algebras and varieties of semilattice modes, and described the semirings determining their semi-linearizations, the varieties of semimodules having these algebras as idempotent subreducts. The third part is devoted to varieties of differential groupoids and more general differential modes, and provides the semirings of the semi-linearizations of these varieties. This paper is a direct continuation of the first and second parts appearing with the same title (4) and (5). In the first part, we considered the problem of constructing a (commutative unital) semiring defining the variety of semimodules whose idempotent subreducts lie in a given variety of modes, and such that each semimodule-embeddable member of this mode variety embeds into a semimodule over the constructed semiring. We described a general construction of such semirings, with basic examples and some general properties. In the second part, we investigated selected varieties of modes, and described the semirings determining varieties of semimodules having algebras of these classes as subreducts, and discussed properties of the corresponding semi-affine spaces. In particular, we investigated varieties of affine spaces, varieties of barycentric algebras and varieties of semilattice modes. The third part is devoted to varieties of differential groupoids and more general

The paper discusses a representation of modes (idempotent and entropic algebras) as subalgebras of so-called functorial sums of cancellative algebras. We show that each mode that has a homomorphism onto an algebra satisfying a certain additional condition, with corresponding cancellative congruence classes, embeds into a functorial sum of cancellative algebras. We also discuss typical applications of this result.

We establish a duality between distributive bisemilattices and certain compact left normal bands. The main technique in the proof utilizes the idea of Plonka sums.

A significant contribution to the analysis of certain aspects of the communicating processes model was made by D. Benson’s proposal to view incompletely specified nondeterministic processes as modules over certain semirings, and dually as comodules over corresponding coalgebras. The effectiveness of the proposal in treating the synthesis of such processes under mutual communication depended on the good behaviour of these algebraic systems with respect to tensor products. The aim of the paper is to draw attension to the algebraic theory underlying Benson’s proposal, the theory of entropic algebras. Working with entropic algebras guarantees that tensor products are sufficiently well-behaved to make Benson’s theory work.