Rickart residuated lattices

Rasouli, Saeed

doi:10.1007/s00500-021-06227-w

Cited by 10 publications

(6 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This section deals with the pure spectrum of an mp-residuated lattice. For the basic facts concerning pure filters of a residuated lattice, the reader is referred to [28].…”

Section: The Pure Spectrum Of An Mp-residuated Latticementioning

confidence: 99%

Mp-Residuated Lattices

RASOULI,

DEHGHANI

2024

KgJMat

View full text Add to dashboard Cite

This paper is devoted to the study of a fascinating class of residuated lattices, the so-called mp-residuated lattice, in which any prime ﬁlter contains a unique minimal prime ﬁlter. A combination of algebraic and topological methods is applied to obtain new and structural results on mp-residuated lattices. It is demonstrated that mp-residuated lattices are strongly tied up with the dual hull-kernel topology. Especially, it is shown that a residuated lattice is mp if and only if its minimal prime spectrum, equipped with the dual hull-kernel topology, is Hausdorﬀ if and only if its prime spectrum, equipped with the dual hull-kernel topology, is normal. The class of mp-residuated lattices is characterized by means of pure ﬁlters. It is shown that a residuated lattice is mp if and only if its pure ﬁlters are precisely its minimal prime ﬁlters, if and only if its pure spectrum is homeomorphic to its minimal prime spectrum, equipped with the dual hull-kernel topology.

show abstract

“…This section deals with the pure spectrum of an mp-residuated lattice. For the basic facts concerning pure filters of a residuated lattice, the reader is referred to [28].…”

Section: The Pure Spectrum Of An Mp-residuated Latticementioning

confidence: 99%

Mp-Residuated Lattices

RASOULI,

DEHGHANI

2024

KgJMat

View full text Add to dashboard Cite

show abstract

“…In this section we shall define the P P -quantales as an abstraction of P P -rings (= Baer rings) [1], [31], [43], Stone lattices [7], [43], Stone M V -algebras [9], Baer BL -algebras [32], Stone residuated lattices [36], [41],etc.…”

Section: P P -Quantalesmentioning

confidence: 99%

“…We mention that most of the results of this paper extend some theorems that appear in the case of commutative rings ( [1], [3], [5], [6], [11], [12], [30], [35], [43], [48], [49]), bounded distributive lattices ( [4], [17], [18], [26], [30], [40], [43], [45]), MV-algebras ( [9], [14]), BL -algebras [32], residuated lattices ( [36], [41]),etc.…”

Section: Introductionmentioning

confidence: 99%

Reticulation of a quantale, pure elements and new transfer properties

Georgescu¹

2020

Preprint

View full text Add to dashboard Cite

We know from a previous paper that the reticulation of a coherent quantale A is a bounded distributive lattice L(A) whose prime spectrum is homeomorphic to m -prime spectrum of A. In this paper we shall prove several results on the pure elements of the quantale A by means of the reticulation L(A). We shall investigate how the properties of σ -ideals of L(A) can be transferred to pure elements of A. Then the pure elements of A are used to obtain new properties and characterization theorems for some important classes of quantales: normal quantales, mp -quantales, P F -quantales, purified quantales and P P -quantales.

show abstract

“…Therefore, the notion of purity is investigated, and as an outcome, various interesting algebraic and topological results are discovered. Although, this concept in residuated lattices has been reviewed by Rasouli (2021), however, here we present some additional properties for this concept that seem to shed more light on algebraic and topological situations. In this paper, we study the purely-prime filters of a residuated lattice, and obtain various new and engaging results.…”

Section: Introductionmentioning

confidence: 95%

The pure spectrum of a residuated lattice

Rasouli¹,

Dehghani²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

This paper studies a fascinating type of filter in residuated lattices, the so-called pure filters. A combination of algebraic and topological methods on the pure filters of a residuated lattice is applied to obtain some new and structural results. The notion of purely-prime filters of a residuated lattice has been investigated, and a Cohen-type theorem has been obtained. It is shown that the pure spectrum of a residuated lattice is a compact sober space, and a Grothendieck-type theorem has been demonstrated. It is proved that the pure spectrum of a Gelfand residuated lattice is a Hausdorff space, and deduced that the pure spectrum of a Gelfand residuated lattice is homeomorphic to its usual maximal spectrum. Finally, the pure spectrum of an mp-residuated lattice is investigated and verified that a given residuated lattice is mp iff its minimal prime spectrum is equipped with the induced dual hull-kernel topology, and its pure spectrum is the same. 1

show abstract

Rickart residuated lattices

Cited by 10 publications

References 27 publications

Mp-Residuated Lattices

Mp-Residuated Lattices

Reticulation of a quantale, pure elements and new transfer properties

The pure spectrum of a residuated lattice

Contact Info

Product

Resources

About