On extensions of monotone commutative monoids on bounded lattices
by
Susanne Saminger-Platz
Department of Knowledge-Based Mathematical Systems, Johannes Kepler University Linz, Austria
Coauthors: R. Mesiar, E.P. Klement

Many-valued logics are usually based on a bounded lattice (L, ≤ , 0, 1) of truth values not necessarily forming a chain. In such a case, the conjunction is interpreted by a monotone commutative associative operation T on L whose identity is the top element of the corresponding lattice. Such operations are also referred to as triangular norms or, briefly, t-norms. Clearly, the structure of the lattice L influences which and how many t-norms can be defined on L. In our contribution we discuss the problem of extending a t-norm T^S acting on a bounded sublattice S to a t-norm T on L. Inspired by ideas of A.H. Clifford (in the context of ordinal sums of abstract semigroups) and by results on ordinal sums of t-norms on the unit interval we propose an extension of T^S on S to L, denoted by T^L_T^S, which, in case it yields t-norm, is the strongest possible t-norm extension. We will clarify under which conditions T^L_T^S is indeed a t-norm for arbitrary T^S and arbitrary bounded sublattice S.

Date received: May 9, 2008