Let \(\text{Map}(S,T)\) denote the set of all functions from \(S\) to \(T\). Let \(J\) be a (complete) bounded linearly ordered lattice with involution.
The authors consider the family \(X= \text{Map}(S, \text{Map}(J, [0,1]))\). The operations on \(X\) come pointwise from the operations available on the set \(M= \text{Map}(J, [0,1])\).
A standard method to equip \(\text{Map}(J, [0,1 ])\) with interesting operations is given by taking the “convolution” of the involutive lattice operations of \(J\) with respect to the corresponding operations on the real unit inveral \([0,1]\). The authors endow \(M\) with such operations and develop their algebraic properties. Subalgebras of \(M\) are considered in the introductory sections. Criteria are given for these subalgebras to satisfy various kinds of lattice-theoretical axioms. In the sixth section, the special case \(J= [0,1]\) is considered, and notions of t-norms and t-conorms are examined. The authors close with a number of questions and problems.