In this paper we introduce the concept of an -representation of an algebra which is a common generalization of subdirect, full subdirect and weak direct representation of . Here we characterize such representations in terms of congruence relations.
Let be a type of algebras. A valuation of terms of type is a function assigning to each term of type a value . For , an identity of type is said to be -normal (with respect to valuation ) if either or both and have value . Taking with respect to the usual depth valuation of terms gives the well-known property of normality of identities. A variety is called -normal (with respect to the valuation ) if all its identities are -normal. For any variety , there is a least...
A duality between -ary varieties and -ary algebraic theories is proved as a direct generalization of the finitary case studied by the first author, F.W. Lawvere and J. Rosick’y. We also prove that for every uncountable cardinal , whenever -small products commute with -colimits in , then must be a -filtered category. We nevertheless introduce the concept of -sifted colimits so that morphisms between -ary varieties (defined to be -ary, regular right adjoints) are precisely the functors...
We establish categorical dualities between varieties of isotropic median algebras and suitable categories of operational and relational topological structures. We follow a general duality theory of B.A. Davey and H. Werner. The duality results are used to describe free isotropic median algebras. If the number of free generators is less than five, the description is detailed.
An alternative (tree-based) semantics for a class of regular expressions is proposed that assigns a central rôle to the + operator and thus to nondeterminism and nondeterministic choice. For the new semantics a consistent and complete axiomatization is obtained from the original axiomatization of regular expressions by Salomaa and by Kozen by dropping the idempotence law for + and the distribution law of • over +.
We define semantically a partial multiplication on the lattice of all e–varieties of regular semigroups. In the case that the first factor is an e–variety of orthodox semigroups we describe our multiplication syntactically in terms of biinvariant congruences.