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Peter Jipsen (Chapman University, USA)

Residuated semigroups, partially ordered algebras and ±-preclones. 

Abstract

Similar to the construction of Clifford semigroups, we show how to construct unions of residuated semigroups, as well as how to decompose them into disjoint fibers (joint work with S. Bonzio, J. Gil-Ferez, A. Prenosil, M. Sugimoto). The general theory of po-algebras with operations that are order-preserving or order-reversing in each argument leads to the concept of ±-preclones that classify such po-algebras up to ±-term equivalence. In collaboration with E. Lehtonen and R. Pöschel, we develop a Galois connection between S-preclones and S-relational clones for a monoid S, where S = {+,-} is the special case of interest here, and use it to find the maximal ±-preclones and minimal ±-preclones over a 2-element set. If time permits, we conclude with the result that residuated lattice-ordered monoids do not have the amalgamation property (with S. Santschi).

Stefano Bonzio & Gavin St. John (Università di Cagliari)

McCarthy algebras and their extensions.

Abstract

The variety if McCarthy algebras plays the role of algebraic semantics for McCarthy logic, introduced by McCarthy in ’59 to reason with partial predicates and failures in computation. In the first part of the talk, we provide an axiomatization for McCarthy algebras (solving an open problem by Konikowska) and a semilattice decomposition theorem for every algebra in the variety.

In the second part, we expose the preliminary results of an ongoing work (with R. Giuntini and F. Paoli) on the extensions of McCarthy algebras with a semilattice disjunction, particularly useful for process algebras with conditional guards and a construct for faulty programs.