It has long been known in universal algebra that any distributive sublattice of congruences of an algebra which consists entirely of commuting congruences yields a sheaf representation of the algebra. In this paper we provide a generalisation of this fact and prove a converse of the generalisation. To be precise, we exhibit a one-to-one correspondence (up to isomorphism) between soft sheaf repr...

Given a presentation of a finitely presented group, there is a natural way to represent the group as the fundamental group of a 2-complex. The first part of this paper demonstrates one possible way to represent a finitely presented algebra S in a similarly compact form. From a presentation of the algebra, we construct a quiver with relations whose path algebra is finite dimensional. When we adj...

We introduce a notion of Hopf-Lie-Rinehart algebra and show that the universal algebra of a Hopf-Lie-Rinehart algebra acquires an ordinary Hopf algebra structure. Subject classification: Primary 16W30, Secondary 16S32 17B35

In this paper we revisit and extend the constructions of Glauberman and Doro on groups with triality and Moufang loops to Hopf algebras. We prove that the universal enveloping algebra of any Lie algebra with triality is a Hopf algebra with triality. This allows us to give a new construction of the universal enveloping algebras of Malcev algebras. Our work relies on the approach of Grishkov and ...

Given a presentation of a finitely presented group, there is a natural way to represent the group as the fundamental group of a 2-complex. The first part of this paper demonstrates one possible way to represent a finitely presented algebra S in a similarly compact form. From a presentation of the algebra, we construct a quiver with relations whose path algebra is finite dimensional. When we adj...

A finite universal algebra is called strictly simple if it is simple and has no nontrivial subalgebras. An algebra is said to be Abelian if for every term t(x,y) and for all elements a, b, c, d, we have the following implication: t(a,c) = t(a,d) —> t(b,c) = t(b,d) . It is shown that every finite simple Abelian universal algebra is strictly simple. This generalizes a well-known fact about Abelia...

A finite universal algebra is called strictly simple if it is simple and has no nontrivial subalgebras. An algebra is said to be Abelian if for every term t(x, ȳ) and for all elements a, b, c̄, d̄, we have the following implication: t(a, c̄) = t(a, d̄) −→ t(b, c̄) = t(b, d̄). It is shown that every finite simple Abelian universal algebra is strictly simple. This generalizes a well known fact about Ab...

Uf(a,ß,Y,x): Universal-logic-module for ternary functions, based upon Reed-Muller expansion, i.e. aodulo-algebra expansion Uh(c,n,x): basic Universal-logic-module for ternary function«, based upon modulo-algebra expansion uh(5,n,x) t ® n x F]: column vector whose entries are the 3 values of f(x), arranged in Increasing order of y, n-1 yI x^ 1-0 LgCK)]: coefficient column vector based upon modul...

In the framework of fuzzy algebras with fuzzy equalities and acomplete lattice as a structure of membership values, we investigate fuzzyequational classes. They consist of special fuzzy algebras fullling the samefuzzy identities, dened with respect to fuzzy equalities. We introduce basicnotions and the corresponding operators of universal algebra: construction offuzzy subalgebras, homomorphisms...