The structure of idempotent residuated chains

Chén, Wěi; Zhao, Xianzhong

doi:10.1007/s10587-009-0031-5

Cited by 11 publications

(6 citation statements)

References 8 publications

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“…Remark 3.16. Similar constructions also using the Green equivalence relation D were considered in [8,7] for idempotent residuated chains and conic idempotent residuated lattices.…”

Section: Nested Sumsmentioning

confidence: 99%

Semilinear idempotent distributive l-monoids

Santschi¹

2022

Preprint

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We prove a representation theorem for totally ordered idempotent monoids via a nested sum construction. Using this representation theorem we obtain a characterization of the subdirectly irreducible members of the variety of semilinear idempotent distributive -monoids and a proof that its subvariety lattice is countably infinite. For the variety of commutative idempotent distributive -monoids we give an explicit description of its subvariety lattice and show that each of its subvarieties is axiomatized relative to it by a single equation. Finally we give a characterization of which spans of totally ordered idempotent monoids have an amalgam in the class of totally ordered monoids, showing in particular that the class of totally ordered commutative idempotent monoids has the strong amalgamation property and that various classes of distributive -monoids do not have the amalgamation property. We also show that exactly seven non-trivial finitely generated subvarieties of the variety of semilinear idempotent distributive -monoids have the amalgamation property; we are able to determine for all but three of its subvarieties whether they have the amalgamation property or not.

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“…Remark 3.16. Similar constructions also using the Green equivalence relation D were considered in [8,7] for idempotent residuated chains and conic idempotent residuated lattices.…”

Section: Nested Sumsmentioning

confidence: 99%

Semilinear idempotent distributive l-monoids

Santschi¹

2022

Preprint

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“…quite widely in the literature (see, e.g., [8,21,6,5,20,15,11,4]), notably for Brouwerian algebras, where the product coincides with the meet, and odd Sugihara monoids, where the product is commutative and the map x Þ Ñ xze is an involution. The monoidal structure of any idempotent residuated lattice A is a unital band and the relation on A defined by a Ď b :ðñ a¨b " a is a preorder that we call the monoidal preorder of A; if the product of A is also commutative, then xA,¨, ey is a unital meet-semilattice with order Ď and greatest element e. When A is totally ordered -that is, A is a residuated chain -the product has the further property that a¨b P ta, bu for all a, b P A; we call residuated lattices satisfying this condition conservative, noting that semigroups with this property are called quasitrivial (see, e.g., [7]).…”

Section: Structural Properties Of Idempotent Residuated Lattices Have Been Studiedmentioning

confidence: 99%

“…The following lemma describes the properties of elements of A that are central and non-central (cf. [5,Proposition 3.1]).…”

Section: Idempotent Residuated Chainsmentioning

confidence: 99%

Structure theorems for idempotent residuated lattices

2020

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In this paper we study structural properties of residuated lattices that are idempotent as monoids. We provide descriptions of the totally ordered members of this class and obtain counting theorems for the number of finite algebras in various subclasses. We also establish the finite embeddability property for certain varieties generated by classes of residuated lattices that are conservative in the sense that monoid multiplication always yields one of its arguments. We then make use of a more symmetric version of Raftery's characterization theorem for totally ordered commutative idempotent residuated lattices to prove that the variety generated by this class has the amalgamation property. Finally, we address an open problem in the literature by giving an example of a noncommutative variety of idempotent residuated lattices that has the amalgamation property.

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“…Among them, semiconic ones make a valuable contribution, because they include several important algebraic counterparts of substructural logics (see [19]). Recently, algebra properties for semiconic CRLs have been given by many authors ( see [4,5,[7][8][9][10][12][13][14][15][16][17][18][19][20][21]). In [20], the authors obtain a structure theorem for semilinear idempotent CRLs.…”

Section: Introductionmentioning

confidence: 99%

“…By Lemma 3(5), (L * , ≤) is a totally ordered set, which implies thatL * is a sublattice of L. Let a * , b * ∈ L * . If a * , b * ≤ e, then a * b * = a * ∧ b * ∈ L * .…”

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confidence: 97%

The Structure of Semiconic Idempotent Commutative Residuated Lattices

Chen

2023

Preprint

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In this paper, we study semiconic idempotent commutative residuated lattices. After giving some properties of such residuated lattices, we obtain a structure theorem for semiconic idempotent com- mutative residuated lattices. As an application, we make use of the structure theorem to prove that the variety of strongly semiconic idempotent commutative residuated lattices has the amalgamation property.

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The structure of idempotent residuated chains

Cited by 11 publications

References 8 publications

Semilinear idempotent distributive l-monoids

Semilinear idempotent distributive l-monoids

Structure theorems for idempotent residuated lattices

The Structure of Semiconic Idempotent Commutative Residuated Lattices

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