Alden F. Pixley

Polynomial completeness in algebraic systems

ALGEBRAS, LATTICES, AND VARIETIES Algebras, Languages, Clones, Varieties Congruence Properties CHARACTERIZATIONS OF EQUIVALENCE LATTICES Introduction Arithmeticity Compatible Function Lifting PRIMALITY AND GENERALIZATIONS Primality and Functional Completeness Near Unanimity Varieties Arithmetical Varieties Generalizations of Primality Categorical Equivalence AFFINE COMPLETE VARIETIES Introduction and Instructive Examples General properties Varieties with a Finite Residual Bound Locally Finite Affine Complete Varieties POLYNOMIAL COMPLETENESS IN SPECIAL VARIETIES Strictly Locally Affine Complete Algebras Modules Lattices Algebras Based on Distributive Lattices Semilattices Miscellaneous Results

Functional and Affine Completeness and Arithmetical Varieties

Our understanding of affine complete and functionally complete algebras and varieties of affine complete algebras is closely related to our understanding of arithmetical algebras and varieties. Because of this we first survey basic results about arithmetical algebras and varieties, emphasizing finite algebras and finitely generated varieties. The main outlines of the theory of congruence distributive affine complete varieties are discussed and then attention is focused on arithmetical affine complete varieties, emphasizing some recent results describing their structure. In particular we shall examine properties of finite, arithmetical, affine complete algebras having no proper subalgebras (FACS algebras) since any affine complete arithmetical variety of finite type is generated by such an algebra. We shall present some interesting sufficient conditions for a FACS algebra to generate an arithmetical (and hence affine complete) variety.