Pałasińska and Pigozzi developed a theory of partially ordered varieties and quasi-varieties of algebras with the goal of addressing issues pertaining to the theory of algebraizability of logics involving an abstract form of the connective of logical implication. Following their lead, the author has abstracted the theory to cover the case of algebraic systems, systems that replace algebras in t...

We introduce a new class of noncommutative rings-Galois orders, realized as certain subrings of invariants in skew semigroup rings, and develop their structure theory. The class of Galois orders generalizes classical orders in noncommutative rings and contains many classical objects, such as the Generalized Weyl algebras, the universal enveloping algebra of the general linear Lie algebra, assoc...

As it is well known, the generalization of universal-algebraic notions to partial algebras yields different kinds of homomorphisms, subalgebras, validity of equations etc. Three types of equations: existential, weak and strong, have been investigated by many algebraists with interesting results on algebraic characterization of equationally definable classes of partial algebras. One of the most ...

We present sufficient conditions under which effective descent morphisms in a quasivariety of universal algebras are the same as regular epimorphisms and examples for which they are the same as regular epimorphisms satisfying projectivity. 1. Preliminaries A variety is a full subcategory of the category of structures for a first order (one sorted) language, closed under substructures, products ...

We introduce some new algebraic structures arising naturally in the geometry of CY manifolds and mirror symmetry. We give a universal construction of CY algebras in terms of a noncommutative symplectic DG algebra resolution. In dimension 3, the resolution is determined by a noncommutative potential. Representation varieties of the CY algebra are intimately related to the set of critical points,...

Every ring R with identity satisfies the following property: the factor ideals of R (i.e., those ideals I such that I+ J= R and In J= (0) for some ideal J) form a Boolean sublattice of the lattice of all ideals of R. The universal algebraic abstraction of this property is known as Boolean factor congruences (BFC) or as the strict refinement property; more examples of algebras having BFC are lat...

We define varieties of algebras for an arbitrary endofunctor on a cocomplete category using pairs of natural transformations. This approach is proved to be equivalent to one of equational classes defined by equation arrows. Free algebras in the varieties are investigated and their existence is proved under the assumptions of accessibility. In universal algebra we deal with varieties – classes o...