New results on topological effect algebras

Abstract

In this paper, by considering the notion of effect algebra and by using of a new ideal in an effect algebra E, we construct a topology τ on E, and we show that (E,τ) is a topological effect algebra. Then we obtain some conditions under which that (E,τ) is a Hausdorff space. Also, we obtain some results about connected components of this topological space, and we construct a quotient topological effect algebra.

Primary decomposition of A-ideals in MV-semimodules

“…Lemma 2.7. [14] Let I ∈ I(M ), S ⊆ M be ∧-closed and S ∩ I = ∅. Then there exists a maximal ideal P of M such that P ⊇ I and P ∩ S = ∅.…”

“…By presentation of definition of M V -modules, some researches were encouraged for working on M V -modules and related structures (see, for instance, [2,15]). Specially, some researches obtained some results on ideals and decomposition of ideals in M V -algebras and M V -modules (see, for instance, [9,12,13,14]). In recent years, Di Nola et al studied the notions of M V -semiring and semimodules and investigated related results [7,8,11].…”

“…Since M V -semirings are a special class of semirings, we hope that new definition of M V -modules helps us to explain M V -modules better than the last. For example, in [14], we introduced the concept of primary decomposition of A-ideals in M V -modules that it was not easy. In this paper, we study and present definitions of primary ideals of M V -semirings, primary Aideals of M V -semimodules, and primary (reduced primary) decomposition of A-ideals in M Vsemimodules.…”

\(L\)-Modules

“…L-Algebras under conditions can be equivalent to other algebras such as BCK-algebras, M V -algebras, etc. Considering that we have spent a relatively large amount of time studying modular structures (for instance, see [3,4,9,10,11,12,13]), in order to complete and consolidate our study in this field, we have decided to define L-modules as an action of an L-algebra over an Abelian group. We hope that this definition can help us to clarify the structure of L-algebras.…”