Abstract. We study direct product representations of algebras in varieties. We collect several conditions expressing that these representations are definable in a first-orderlogic sense, among them the concept of Definable Factor Congruences (DFC). The main results are that DFC is a Mal'cev property and that it is equivalent to all other conditions formulated; in particular we prove that V has DFC if and only if V has 0 & 1 and Boolean Factor Congruences. We also obtain an explicit first order definition Φ of the kernel of the canonical projections via the terms associated to the Mal'cev condition for DFC, in such a manner it is preserved by taking direct products and direct factors. The main tool is the use of central elements, which are a generalization of both central idempotent elements in rings with identity and neutral complemented elements in a bounded lattice.
Let A be an algebra. We say that the functions f1, . . . , fm : A n → A are algebraic on A provided there is a finite system of term-equalities t k (x, z) = s k (x, z) satisfying that for each a ∈ A n , the m-tuple (f 1 (a), . . . , f m (a)) is the unique solution in A m to the system t k (a, z) = s k (a, z). In this work we present a collection of general tools for the study of algebraic functions, and apply them to obtain characterizations for algebraic functions on distributive lattices, Stone algebras, finite abelian groups and vector spaces, among other well known algebraic structures.
Let V be a discriminator variety such that the class B=[A # V: A is simple and has no trivial subalgebra] is closed under ultraproducts. This property holds, for example, if V is locally finite or if the language is finite. Let v(V) and q(V) denote the lattice of subvarieties and subquasivarieties of V, respectively. We prove that q(V) is modular iff q(V) is distributive iff v(V) satisfies a certain condition where the case in which the language has a constant symbol is``v(V) is a chain or q(V) =v(V).'' We give an isomorphism between q(V) and a lattice constructed in terms of v(V). Via this isomorphism we characterize the completely meet irreducible (prime) elements of q(V) in terms of the completely meet irreducible elements of v(V). We conclude the paper with applications to the varieties of Boolean algebras, relatively complemented distributive lattices, 4ukasiewicz algebras, Post algebras, complementary semigroups of rank k (x n rx)-rings, R 5 lattices (P-algebras, B-algebras), and monadic algebras.
Academic Press
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.