Applications of finite duality to locally finite varieties of BL-algebras. (English)
Artemov, Sergei (ed.) et al., Logical foundations of computer science. International symposium, LFCS 2009, Deerfield Beach, FL, USA, January 3‒6, 2009. Proceedings. Berlin: Springer (ISBN 978-3-540-92686-3/pbk). Lecture Notes in Computer Science 5407, 1-15 (2009).
Summary: We are concerned with the subvariety of commutative, bounded, and integral residuated lattices, satisfying divisibility and prelinearity, namely, BL-algebras. We give an explicit combinatorial description of the category that is dual to finite BL-algebras. Building on this, we obtain detailed structural information on the locally finite subvarieties of BL-algebras that are analogous to Grigolia’s subvarieties of finite-valued MV-algebras. As an illustration of the power of the finite duality presented here, we give an exact recursive formula for the cardinality of free finitely generated algebras in such varieties.