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Quantum quasigroups and quantum loops are self-dual objects providing a general framework for the nonassociative extension of quantum group techniques. This paper examines their one-sided analogues, which are not self-dual. Just as quantum quasigroups are the "quantum" version of quasigroups, so one-sided quantum quasigroups are the "quantum" version of left or right quasigroups. [ABSTRACT FROM AUTHOR]

Quantum quasigroups and quantum loops are self-dual objects providing a general framework for the nonassociative extension of quantum group techniques. This paper examines their one-sided analogues, which are not self-dual. Just as quantum quasigroups are the 'quantum' version of quasigroups, so one-sided quantum quasigroups are the 'quantum' version of left or right quasigroups. [ABSTRACT FROM AUTHOR]

Following the prototype of dimonoids, directional algebras are obtained from universal algebras by splitting each fundamental operation into a number of distinct fundamental operations corresponding to directions or selected arguments in the original fundamental operation. Thus dimonoids are directional semigroups, with left- and right-directed multiplications. Directional quasigroups appear in a number of versions, depending on the axiomatization chosen for quasigroups, but this paper concentrates on 4-diquasigroups, which incorporate a left and right quasigroup structure. While introducing several new instances of 4-diquasigroups, including dicores and group-representable diquasigroups, the paper is primarily devoted to the study of undirected replicas of directional binary algebras, dimonoids, digroups, and diquasigroups, where the two directed multiplications are identified. Undirected replicas of diquasigroups are two-sided quasigroups, and thus offer a new approach to the construction of quasigroups of various kinds. [ABSTRACT FROM AUTHOR]

Following the prototype of dimonoids, diquasigroups are directed versions of quasigroups, where the structure is split into left and right quasigroups on the same set. The linear and affine diquasigroups that form the topic of this paper are built on the foundation of a module. In this context, various issues that may be difficult to handle in the general case, for example identification of the largest two-sided quasigroup image, become more tractable. An appropriate universal algebraic language for affine diquasigroups is established, and the entropic models of this language are characterized. Various interesting classes of linear and affine diquasigroups are singled out for special attention, such as internally associative, Bol, and symmetric diquasigroups. The problem of determining which linear diquasigroups have an abelian group as their undirected replica is raised. One sufficient condition is provided, formulated in terms of a differential calculus for one-sided quasigroup words. [ABSTRACT FROM AUTHOR]

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 analogs are dyadic intervals. Algebraically, dyadic polygons carry the structure of a commutative, entropic and idempotent algebra under the binary operation of arithmetic mean. In this paper, dyadic intervals and triangles are classified to within affine or algebraic isomorphism, and dyadic polygons are shown to be finitely generated as algebras. The auxiliary results include a form of Pythagoras' theorem for dyadic affine geometry. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]