Residuation in non-associative MV-algebras

TL;DR

This paper explores how non-associative MV-algebras (NMV-algebras) can be transformed into weaker residuated structures called conditionally residuated posets, extending the algebraic framework beyond traditional MV-algebras.

Contribution

It introduces the concept of converting NMV-algebras into conditionally residuated posets and establishes a two-way correspondence, broadening the understanding of algebraic structures related to MV-algebras.

Findings

NMV-algebras can be converted into conditionally residuated posets.

Every such poset can be organized into an NMV-algebra.

Stronger algebraic structures with monotonic operations can be converted into residuated posets.

Abstract

It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law. In our previous papers we introduced the concept of an NMV-algebra which is a non-associative modification of an MV-algebra. The natural question arises if an NMV-algebra can be converted into a residuated structure, too. Contrary to MV-algebras, NMV-algebras are not based on lattices but only on directed posets and the binary operation need not be associative and hence we cannot expect to obtain a residuated lattice but only an essentially weaker structure called a conditionally residuated poset. Considering several additional natural conditions we show that every NMV-algebra can be converted in such a structure. Also conversely, every such structure can be organized into an NMV-algebra. Further, we study a bit more stronger version of an algebra where the binary operation is even monotonous. We show that such an algebra can be organized into a residuated poset and, conversely, every residuated poset can be converted in this structure.