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arXiv · 2004.14127

Extensions of posets with an antitone involution to residuated structures

Ivan Chajda·Miroslav Kolařík·Helmut Länger
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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.

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Ivan Chajda, Miroslav Kolařík, Helmut Länger. 2020-04-29. Extensions of posets with an antitone involution to residuated structures. https://arxiv.org/abs/2004.14127

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