We show that the universal Horn theory of relational Kleene algebras is Π 1complete.

The purpose of this paper is to introduce a cohomology theory for abelian matched pairs of Hopf algebras and to explore its relationship to Sweedler cohomology, to Singer cohomology and to extension theory. An exact sequence connecting these cohomology theories is obtained for a general abelian matched pair of Hopf algebras, generalizing those of Kac and Masuoka for matched pairs of finite grou...

The category of Hopf monoids over an arbitrary symmetric monoidal category as well as its subcategories of commutative and cocommutative objects respectively are studied, where attention is paid in particular to the following questions: (a) When are the canonical forgetful functors of these categories into the categories of monoids and comonoids respectively part of an adjunction? (b) When are ...

We reduce the basis construction problem for Hopf algebras generated by skew-primitive semi-invariants to a study of special elements, called “super-letters,” which are defined by Shirshov standard words. In this way we show that above Hopf algebras always have sets of PBW-generators (“hard” super-letters). It is shown also that these Hopf algebras having not more than finitely many “hard” supe...

In this paper, we introduce and investigate monadic NM-algebras: a variety of NM-algebras equipped with universal quantifiers. Also, we obtain some conditions under which monadic NM-algebras become monadic Boolean algebras. Besides, we show that the variety of monadic NM-algebras faithfully the axioms on quantifiers in monadic predicate NM logic. Furthermore, we discuss relations between monadi...

Generalized basic logic algebras (GBL-algebras for short) have been introduced in [JT02] as a generalization of Hájek’s BL-algebras, and constitute a bridge between algebraic logic and l-groups. In this paper we investigate normal GBL-algebras, that is, integral GBL-algebras in which every filter is normal. For these structures we prove an analogue of Blok and Ferreirim’s [BF00] ordinal sum dec...

In this paper we will study deformations of A∞-algebras. We will also answer questions relating to Moore algebras which are one of the simplest nontrivial examples of an A∞-algebra. We will compute the Hochschild cohomology of odd Moore algebras and classify them up to a unital weak equivalence. We will construct miniversal deformations of particular Moore algebras and relate them to the univer...

The category of group-graded modules over an abelian group G is a monoidal category. For any bicharacter of G this category becomes a braided monoidal category. We define the notion of a Lie algebra in this category generalizing the concepts of Lie super and Lie color algebras. Our Lie algebras have n-ary multiplications between various graded components. They possess universal enveloping algeb...

In this paper we introduce the concept of Rees closure for subalgebras universal algebras. We denote by △

A category is said to be alg-universal, if every category of universal algebras can be fully embedded into it. We prove here that the category of varieties and interpretations, or in other words, the category of abstract clones and clone homomorphisms, is alg-universal.