A duality for two-sorted lattices

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A duality for two-sorted lattices. / Rivieccio, Umberto; Jung, Achim.

In: Soft Computing, Vol. 25, No. 2, 05.01.2021, p. 851–868.

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@article{1d8da17aebbb469e924e2f7de1e8a54d,
title = "A duality for two-sorted lattices",
abstract = "A series of representation theorems (some of which discovered very recently) present an alternative view of many classes of algebras related to non-classical logics (e.g. bilattices, semi-De Morgan, Nelson and quasi-Nelson algebras) as two-sorted algebras in the sense of many-sorted universal algebra. In all the above-mentioned examples, we are in fact dealing with a pair of lattices related by two meet-preserving maps. We use this insight to develop a Priestley-style duality for such structures, mainly building on the duality for meet-semilattices of G. Bezhanishvili and R. Jansana. Our approach simplifies all the existing dualities for these algebras and is applicable more generally; in particular, we show how it specialises to the class of quasi-Nelson algebras, which has not yet been studied from a duality point of view.",
keywords = "Twist-structure, Nelson algebra, Semi-De Morgan algebra, Priestley duality, Bilattice",
author = "Umberto Rivieccio and Achim Jung",
year = "2021",
month = jan,
day = "5",
doi = "10.1007/s00500-020-05482-7",
language = "English",
volume = "25",
pages = "851–868",
journal = "Soft Computing",
issn = "1432-7643",
publisher = "Springer",
number = "2",

}

RIS

TY - JOUR

T1 - A duality for two-sorted lattices

AU - Rivieccio, Umberto

AU - Jung, Achim

PY - 2021/1/5

Y1 - 2021/1/5

N2 - A series of representation theorems (some of which discovered very recently) present an alternative view of many classes of algebras related to non-classical logics (e.g. bilattices, semi-De Morgan, Nelson and quasi-Nelson algebras) as two-sorted algebras in the sense of many-sorted universal algebra. In all the above-mentioned examples, we are in fact dealing with a pair of lattices related by two meet-preserving maps. We use this insight to develop a Priestley-style duality for such structures, mainly building on the duality for meet-semilattices of G. Bezhanishvili and R. Jansana. Our approach simplifies all the existing dualities for these algebras and is applicable more generally; in particular, we show how it specialises to the class of quasi-Nelson algebras, which has not yet been studied from a duality point of view.

AB - A series of representation theorems (some of which discovered very recently) present an alternative view of many classes of algebras related to non-classical logics (e.g. bilattices, semi-De Morgan, Nelson and quasi-Nelson algebras) as two-sorted algebras in the sense of many-sorted universal algebra. In all the above-mentioned examples, we are in fact dealing with a pair of lattices related by two meet-preserving maps. We use this insight to develop a Priestley-style duality for such structures, mainly building on the duality for meet-semilattices of G. Bezhanishvili and R. Jansana. Our approach simplifies all the existing dualities for these algebras and is applicable more generally; in particular, we show how it specialises to the class of quasi-Nelson algebras, which has not yet been studied from a duality point of view.

KW - Twist-structure

KW - Nelson algebra

KW - Semi-De Morgan algebra

KW - Priestley duality

KW - Bilattice

U2 - 10.1007/s00500-020-05482-7

DO - 10.1007/s00500-020-05482-7

M3 - Article

VL - 25

SP - 851

EP - 868

JO - Soft Computing

JF - Soft Computing

SN - 1432-7643

IS - 2

ER -