Recent zbMATH articles in MSC 06D https://www.zbmath.org/atom/cc/06D 2021-04-16T16:22:00+00:00 Werkzeug Three-way convex systems and three-way fuzzy convex systems. https://www.zbmath.org/1456.68210 2021-04-16T16:22:00+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.hailong Summary: 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/1456.06010 2021-04-16T16:22:00+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.arsham The 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/1456.06011 2021-04-16T16:22:00+00:00 "Battenfeld, Ingo" https://www.zbmath.org/authors/?q=ai:battenfeld.ingo Summary: 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/1456.06012 2021-04-16T16:22:00+00:00 "Çaylı, Gül Deniz" https://www.zbmath.org/authors/?q=ai:cayli.gul-deniz Summary: 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.