A survey about compatible operations in some varieties

HERNÁN JAVIER SAN MARTÍN

Petrópolis

Conferencia; EBL (Brazilian Logic Conference); 2014

LNCC (Laboratorio Nacional de Computación Científica)

Compatibility of functions is a classical topic in Universal Algebra related to the notion  of affine completeness. In algebraic logic it is concerned with the possibility of implicitly defining new connectives. In this work we give characterizations of compatible operations in a variety of algebras that properly includes commutative residuated lattices and some generalizations of Heyting algebras. The wider variety considered is obtained by weakening the main characters of residuated lattices $(A, inf,  sup, prod, imp, e)$ but retaining most of their algebraic consequences, and their algebras have  a commutative monoidal structure. The order-extension principle $a leq b$ if and only if $a imp  b geq e$ is replaced by the condition: if $a leq b$, then $a imp b geq e$. The residuation property $c leq a imp b$ if and only if $a prod c leq b$ is replaced by the conditions: if $c leq a imp b$ then $a prod c leq b$, and if $a prod c leq b$ then $e leq a imp b$.