On the algebra of order extended abstract

Abstract

Algebras whose carriers are partially ordered sets and operations are monotone and algebras whose carriers are complete partial orders and operations are continuous are studied. A quotient construction is provided for both types of algebras. The notion of a variety of algebras is defined and it is shown that the analogue of Birkhoff variety theorem holds for ordered algebras but not for continuous algebras. The results presented are a good first step towards a theory of ordered data types and a study of families of interpretations of schemas.