Varieties of Partially ordered algebras

The recent development of abstract algebraic logic (AAL) has led to a reconsideration of the universal algebraic theory of ordered algebras from a new perspective. The familiar equational logic of Birkhoff can be naturally viewed as one of the deductive systems that constitute the main object of study in AAL; technically it is a deductive system of dimension two. Large parts of universal algebra turn out to fit smoothly within the matrix semantics of this deductive system. Ordered algebras in turn can be viewed as special kinds of matrix models of other, closely related, 2-dimensional deductive systems. We consider here a more general notion of ordered algebra in which some operations may be anti-monotone in some arguments; this leads to many different ordered equational logics over the same language. These lectures will be a introduction to universal ordered algebra from this new perspective. Financiamento POCTI - Projecto Reticulados, Álgebra Universal e Lógica Algébrica (CAUL)