Volume 44, Issue 4

Weak relative pseudocomplementation on a meet semilattice S is a partial operation * which associates with every pair ( x, y ) of elements, where x ≥ y , an element z (the weak pseudocomplement of x relative to y ) which is the greatest among elements u such that y = u Λ x . The element z coincides with the pseudocomplement of x in the upper section [ y ) and, if S is modular, with the pseudocomplement of x relative to y . A weakly relatively pseudomented semilattice is said to be extended, if it is equipped with a total binary operation extending *. We study congruence properties of the variety of such semilattices and review some of its subvarieties already described in the literature.

The class of skew lattices can be seen as an algebraic category. It models an algebraic theory in the category of sets where the Green’s relation D is a congruence describing an adjunction to the category of lattices. In this paper we will discuss the relevance of this approach, revisit some known decompositions and relate the order structure of a skew lattice with its coset structure that describes the internal coset decomposition of the respective skew lattice.

We characterize semirings which can be represented by an algebra of binary polynomials of the form a·x + y where the operations are compositions of functions. Furthermore, we classify, which algebras with two binary and two nullary operations (satisfying some natural identities) can be represented in this way, and how these algebras are related to semirings.

In this paper we study the semigroup ℐλ∞${\cal I}_\lambda ^\infty $ of injective partial selfmaps almost everywhere the identity of a set of infinite cardinality λ . We describe the Green relations on ℐλ∞${\cal I}_\lambda ^\infty $, all (two-sided) ideals and all congruences of the semigroup ℐλ∞${\cal I}_\lambda ^\infty $. We prove that every Hausdorff hereditary Baire topology τ on ℐλ∞${\cal I}_\lambda ^\infty $ such that (ℐλ∞${\cal I}_\lambda ^\infty $, τ ) is a semitopological semigroup is discrete and describe the closure of the discrete semigroup ℐλ∞${\cal I}_\lambda ^\infty $ in a topological semigroup. Also we show that for an infinite cardinal λ the discrete semigroup ℐλ∞${\cal I}_\lambda ^\infty $ does not embed into a compact topological semigroup and construct two non-discrete Hausdorff topologies turning ℐλ∞${\cal I}_\lambda ^\infty $ into a topological inverse semigroup.

In 2009 K. Denecke and J. Koppitz proved that for a monoid ℳ of hypersubstitutions M -solid positive varieties of tree languages correspond to M -solid ordered pseudovarieties. In this paper, we will characterize M -solid ordered pseudovarieties in a similar way in which in [14] M -solid varieties, in [3] M -solid quasivarieties, in [11] M -solid pseudovarieties and in [12] M -solid algebraic systems were characterized. The main idea is to show, that we have two Galois-connections and a conjugate pair of additive closure operators. Then we can apply the general theory of conjugate pairs of additive closure operators.

This paper is mainly dedicated to the description of coregular subsemigroups of the symmetric semigroup T n of transformations on an n -element set. Namely, we characterize all coregular transformation semigroups S with | S | ≤ 3. In the subsemigroup E n of all extensive transformations, the coregular elements coincide with the idempotent ones. We characterize all bands within E n . Within the subsemigroup OE n of all order-preserving extensive transformations, we also determine the maximal bands (with respect to the inclusion).

Our purposes in this work include the following: (1) Extend and expand earlier work on symmetric spaces, particularly that done from a nonassociative algebra point of view, from the finite-dimensional setting to the Banach space setting. (2) Take a careful look at the equivalence of the categories of smooth pointed reflection quasigroups (a special class of symmetric spaces) and uniquely 2-divisible Bruck loops (= K -loops = gyrocommutative gyrogroups). (3) Propose a loop-theoretic analog of topological vector spaces. (4) Derive algebraic consequences and equivalences of smoothness notions, particularly the notion of parallel transport. (5) Illustrate the effective interaction of the algebraic operations of reflection, Bruck addition, and coaddition in the test case of parallelograms in symmetric spaces.

The first part of this paper specified the semi-affinization semiring of a mode variety as the universal scalar semiring for semimodules whose idempotent subreducts lie in the given variety of modes. The current part of the paper focusses on some selected varieties of modes (affine spaces, barycentric algebras, semilattice modes), and computes the semi-affinization semirings of these varieties.

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 universal algebra, homomorphisms are usually considered between algebras of the same similarity type. Different from that, the notion of a weak homomorphism, as introduced by E. Marczewski in 1961, does not depend on a signature, but only on the clones of term operations generated by the examined algebras. We generalize this idea by defining weak homomorphisms between F 1 - and F 2 -algebras, where F 1 and F 2 denote not necessarily equal endofunctors of the category of sets. The aim is to show that, in many respects, weak homomorphisms behave very similarly to proper homomorphisms–without restricting the scope of considerations by the necessity of a common type. For instance, concerning a set ℱ of Set -endofunctors that weakly preserve kernels, the class of all algebras of types from ℱ equipped with the class of all weak homomorphisms between these algebras forms a category which admits a canonical factorization structure for morphisms. Furthermore, we treat two product constructions from which the notion of a weak homomorphism naturally arises.

In this paper it will be proved that if a clone C is category equivalent to a centralizer clone then C is a centralizer clone.

While homology theory of associative structures, such as groups and rings, has been extensively studied in the past beginning with the work of Hopf, Eilenberg, and Hochschild, homology of non-associative distributive structures, such as quandles, were neglected until recently. Distributive structures have been studied for a long time. In 1880, C.S. Peirce emphasized the importance of (right) self-distributivity in algebraic structures. However, homology for these universal algebras was introduced only sixteen years ago by Fenn, Rourke, and Sanderson. We develop this theory in the historical context and propose a general framework to study homology of distributive structures. We illustrate the theory by computing some examples of 1-term and 2-term homology, and then discussing 4-term homology for Boolean algebras. We outline potential relations to Khovanov homology, via the Yang-Baxter operator.