Łukasiewicz-Moisil relation algebras
Let \(n\) be a fixed finite positive integer. An \textsl{\(n\)-valued Łukasiewicz-Moisil algebra} is a bounded distributive lattice together with a unary operation that satisfies the De~Morgan law and \(n\) unary operations that are lattice endomorphisms and satisfy several additional postulates. Łukasiewicz-Moisil algebras generalize Boolean algebras. They are algebraic counterparts of Łukasiewicz's multi-valued logics, just as Boolean algebras correspond to 2-valued logic. A relation algebra is, by definition, a Boolean algebra with some additional operations that satisfy a few postulates. By replacing ``Boolean algebra'' in this definition with ``Łukasiewicz-Moisil algebra'', one obtains the definition of ``Łukasiewicz-Moisil relation algebra.'' One may also similarly generalize the definition of \(\text{Rel}(X)\) (the relation algebra of all binary relations on the set \(X\)), obtaining the concrete Łukasiewicz-Moisil relation algebra \(\text{MRel}(X)\). From any ternary relation \(T\) one can construct the complex algebra \(\text{CA}(T)\), which will be a relation algebra if \(T\) has suitable properties. This construction can be similarly generalized, resulting in \(\text{MCA}(T)\), called the \textsl{many-valued complex algebra} of \(T\). Several theorems from the elementary theory of relation algebras still hold in this new setting. For example, a Łukasiewicz-Moisil relation algebra is simple iff it if directly (or subdirectly) indecomposable, and every Łukasiewicz-Moisil relation algebra is isomorphic to a subalgebra of a complex algebra of a particular ternary relation on the ultrafilters of the underlying Łukasiewicz-Moisil algebra. Categorical properties of Łukasiewicz-Moisil relation algebras and their connections with Post algebras and with matrix relation algebras are also investigated.
scientific article; zbMATH DE number 5013024
scientific article