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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

For A ≤ B first order structures in a class K, say that A is an epic substructure of B in K if for every C ∈ K and all homomorphisms g, g : B → C, if g and g agree on A, then g = g . We prove that A is an epic substructure of B in a class K closed under ultraproducts if and only if A generates B via operations definable in K with primitive positive formulas. Applying this result we show that a quasivariety of algebras Q with an n-ary near-unanimity term has surjective epimorphisms if and only if SPnPu(Q RSI ) has surjective epimorphisms. It follows that if F is a finite set of finite algebras with a common near-unanimity term, then it is decidable whether the (quasi)variety generated by F has surjective epimorphisms.

An algebraically expandable class is a class of similar algebras axiomatizable by sentences of the form (∀∃! V eq). The problem investigated in this work is that of finding all algebraically expandable classes within a given variety. A complete solution is presented for a number of varieties, including the classes of Boolean algebras, Stone algebras, semilattices, distributive lattices and generalized Kleene algebras. We also study the problem for the case of discriminator varieties, where we prove that there is a lattice isomorphism between the lattice of all algebraically expandable classes of the variety and a certain lattice of subclasses of the simple members of the variety. Finally this connection is applied to calculating the algebraically expandable subclasses of the varieties of monadic algebras and P-algebras.