Absolutely definable varieties of continuous algebras (Q1117960)

Z-continuous algebras are ordered, Z-complete algebras with Z-continuous operations, where Z is a subset system. An absolutely definable variety of these algebras is a variety defined by a set of inequalities between terms which are elements of the absolutely free Z-continuous algebra over a set of generators. An algebraic variety is a variety defined by inequalities between terms involving no join symbol. The paper provides the characterizations of these two types of varieties, under the assumption that each Z-set is directed, by using the notions of weak subalgebra and of Z-completion of a poset with respect to an order- preserving map.