All seven intermediate logics admitting Craig interpolation also admit uniform interpolation; however, although the modal logic K admits both properties, its extension S4 admits only Craig interpolation and not uniform interpolation (see [1] for details and references). Uniform interpolation for a logic may be viewed as a weaker form of quantifier elimination. This idea is exploited in the mono...

Metric algebras are metric variants of Σ-algebras. They are first introduced in the field of universal algebra to deal with algebras equipped with metric structures such as normed vector spaces. Recently a similar notion of quantitative algebra is used in computer science to formulate computational effects of probabilistic programs. In this paper we show that varieties of metric algebras (class...

A variety V has an alg-universal n-expansion if the addition of n nullary operations to algebras from V produces an alg-universal category. It is proved that any semigroup variety V containing a semigroup that is neither an inflation of a completely simple semigroup nor an inflation of a semilattice of groups has an alg-universal 3-expansion. We say that a variety V is var-relatively alg-univer...

We describe the structure of those locally finite varieties whose first order theory is decidable. A variety is a class of universal algebras defined by a set of equations. Such a class is said to be locally finite if every finitely generated member of the class is finite. It turns out that in order for such a variety to have a decidable theory it must decompose into the varietal product of thr...

Let Θ be a variety of algebras. In every Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We consider also a special categorical invariant K Θ (H) of this geometry. The classical algebraic geometry deals with the variety Θ = Com − P of all associative and commutative algebras over the ground field of constants P. An algebra H in this setting is an extension of the g...

The present paper surveys the history of algebraic represetations of complete directed graphs, known in graph theory as tournaments, or equivalently, relational structures with a trichotomous binary relation. Essentially, two kinds of algebraizations of tournaments were studied in the literature: algebras with one binary operation (called groupoids of tournaments) and algebras with two binary o...

De Concini, Kac and Procesi defined a family of subalgebras U + of a quantized universal enveloping algebra Uq(g), associated to the elements of the corresponding Weyl group W . They are deformations of the universal enveloping algebras U(n+ ∩ Adw(n−)) where n± are the nilradicals of a pair of dual Borel subalgebras. Based on results of Gorelik and Joseph and an interpretation of U + as quantiz...

We introduce the notion of a projective hull for subsets of complex projective varieties parallel to the idea of a polynomial hull in affine varieties. With this concept, a generalization of J. Wermer’s classical theorem on the hull of a curve in Cn is established in the projective setting. The projective hull is shown to have interesting properties and is related to various extremal functions ...

A b s t r a c t. We investigate some properties of two varieties of algebras arising from quantum computation quasi-MV algebras and √ ′ quasi-MV algebras first introduced in [13], [12] and tightly connected with fuzzy logic. We establish the finite model property and the congruence extension property for both varieties; we characterize the quasi-MV reducts and subreducts of √ ′ quasiMV algebras...