Recent zbMATH articles in MSC 06Dhttps://www.zbmath.org/atom/cc/06D2021-04-16T16:22:00+00:00WerkzeugThree-way convex systems and three-way fuzzy convex systems.https://www.zbmath.org/1456.682102021-04-16T16:22:00+00:00"Zhang, Shao-Yu"https://www.zbmath.org/authors/?q=ai:zhang.shaoyu"Li, Sheng-Gang"https://www.zbmath.org/authors/?q=ai:li.shenggang"Yang, Hai-Long"https://www.zbmath.org/authors/?q=ai:yang.hailongSummary: In this paper, we study convex systems under the frame of three-way decision theory. Firstly, we give the notion of three-way convex systems. We obtain an equivalence characterization of three-way convex systems. Then, we propose the concept of three-way fuzzy convex systems in term of \(L\)-convex systems. Some related examples are presented. Finally, we establish a one-to-one correspondence between three-way convex systems and three-way fuzzy convex systems.MV-modules of fractions.https://www.zbmath.org/1456.060102021-04-16T16:22:00+00:00"Banivaheb, Hoda"https://www.zbmath.org/authors/?q=ai:banivaheb.hoda"Borumand Saeid, Arsham"https://www.zbmath.org/authors/?q=ai:borumand-saeid.arshamThe paper considers MV-modules over PMV-algebras.
Let \(A\) be a unital PMV-algebra, \(M\) an MV-module over \(A\) and \(P\) be a \(\cdot\)-prime ideal of \(A\). Let \(0_P(A)\) be the intersection of all prime ideals of \(A\) included in \(P\).
Let \(A'\) be a subalgebra of \(A\) admitting \(P\) as a maximal \(\cdot\)-ideal. Then the quotient PMV-algebra \(A'/0_P(A)\) is local and is a kind of localization of \(A\) at \(P\) (although it is not unique).
Moreover we can correspondingly build an A-ideal \(0_P(M)\) of \(M\), so that (Theorem 3.1) the quotient MV-algebra \(M_P=M/0_P(M)\) is a MV-module over \(A'/0_P(A)\). This module is a localization of \(M\) at \(P\) and should provide an analogue of the ring-theoretic module of fractions over a prime ideal.
In Lemma 3.9, the localization construction is made functorial.
In Section 4, there are relations between ideals of \(M\) and ideals of \(M_P\).
Reviewer: Giacomo Lenzi (Fisciano)A domain-theoretic investigation of posets of sub-\(\sigma\)-algebras (extended abstract).https://www.zbmath.org/1456.060112021-04-16T16:22:00+00:00"Battenfeld, Ingo"https://www.zbmath.org/authors/?q=ai:battenfeld.ingoSummary: Given a measurable space \((X,\mathcal{M})\) there is a (Galois) connection between sub-\(\sigma\)-algebras of \(\mathcal{M}\) and equivalence relations on \(X\). On the other hand equivalence relations on \(X\) are closely related to congruences on stochastic relations. In recent work, Doberkat has examined lattice properties of posets of congruences on a stochastic relation and motivated a domain-theoretic investigation of these ordered sets. Here we show that the posets of sub-\(\sigma\)-algebras of a measurable space do not enjoy desired domain-theoretic properties and that our counterexamples can be applied to the set of smooth equivalence relations on an analytic space, thus giving a rather unsatisfactory answer to Doberkat's question.
For the entire collection see [Zbl 1391.03010].Nullnorms on bounded lattices derived from t-norms and t-conorms.https://www.zbmath.org/1456.060122021-04-16T16:22:00+00:00"Çaylı, Gül Deniz"https://www.zbmath.org/authors/?q=ai:cayli.gul-denizSummary: Nullnorms with an annihilator \(a\) in any point of a bounded lattice are generalizations and unifications of t-norms and t-conorms. This study continues to investigate the construction of nullnorms on bounded lattices. We propose some methods to construct nullnorms derived from t-norms and t-conorms on bounded lattices, where some sufficient and necessary conditions on theirs annihilator are required. As a by-product of these constructions, idempotent nullnorms on bounded lattices are obtained. Further, we provide some illustrative examples of the new classes of nullnorms (idempotent nullnorms) on bounded lattices.