Representable biresiduated lattices
A biresiduated lattice is an algebra \({\mathbf A}=\langle A;\cdot ,/,\backslash ,\wedge ,\vee ,1\rangle \) such that \(\langle A;\wedge,\vee \rangle \) is a lattice in which \(1\) is the maximum element, and \( /\) and \(\backslash \) are binary operations which satisfy the following ``residuation'' properties with respect to the lattice order: \(a\cdot c\leq b\) iff \(c\leq a\backslash b\); \(c\cdot a\leq b\) iff \(c\leq b/a\) for all \(a\), \(b\), \(c\) in \(A\). The corresponding class is denoted by \({\mathcal B}\). These algebras were first studied by Krull as abstractions of ideal lattices of rings enriched with the monoid operation of ideal multiplication. The objective of this paper is to axiomatize the class of all the biresiduated lattices that may be represented as subalgebras of products of linearly ordered biresiduated lattices (these are called representable and the corresponding class is denoted by \({\mathcal L}\)). It is shown that \({\mathcal L}\) is axiomatized, relative to \({\mathcal B}\), by the identity \[ (x\backslash y)\vee ([w\cdot (z\backslash ((y\backslash x)\cdot z))]/w)=1 \] or, equivalently, by \[ (x\backslash y)\vee (w/(w/(((y\backslash x)\backslash z)\backslash z)))=1. \] Representable cancellative and complemented biresiduated lattices are also discussed. Finally, connections with lattice-ordered groups are given.
scientific article; zbMATH DE number 1740520
scientific article