Extensions of posets with an antitone involution to residuated structures
Rings and Algebras
2020-04-30 v1
Abstract
We prove that every not necessarily bounded poset P=(P,\le,') with an antitone involution can be extended to a residuated poset E(P)=(E(P),\le,\odot,\rightarrow,1) where x'=x\rightarrow0 for all x\in P. If P is a lattice with an antitone involution then E(P) is a lattice, too. We show that a poset can be extended to a residuated poset by means of a finite chain and that a Boolean algebra (B,\vee,\wedge,',p,q) can be extended to a residuated lattice (Q,\vee,\wedge,\odot,\rightarrow,1) by means of a finite chain in such a way that x\odot y=x\wedge y and x\rightarrow y=x'\vee y for all x,y\in B.
Cite
@article{arxiv.2004.14127,
title = {Extensions of posets with an antitone involution to residuated structures},
author = {Ivan Chajda and Miroslav Kolařík and Helmut Länger},
journal= {arXiv preprint arXiv:2004.14127},
year = {2020}
}