Sequential convergences on MV-algebras without Urysohn's axiom

Year, pages: 2008, 289 - 300

In a previous author's paper, sequential convergences on an \linebreak $MV$-algebra $\mathcal{A}$ have been studied; the Urysohn's axiom was assumed to be valid. The system of all such convergences was denoted by $\Conv \mathcal{A}$. In the present paper we investigate analogous questions without supposing the validity of the Urysohn's axiom; the corresponding system of convergences is denoted by $\conv \mathcal{A}$. Both $\Conv \mathcal{A}$ and $\conv \mathcal{A}$ are partially ordered by the set-theoretical inclusion. We deal with the properties of $\conv \mathcal{A}$ and the relations between $\conv \mathcal{A}$ and $\Conv \mathcal{A}$. We prove that each interval of $\conv \mathcal{A}$ is a distributive lattice. The system $\conv \mathcal{A}$ has the least element, but it does not possess any atom. Hence it is either a singleton set or it is infinite. We consider also the relations between $\conv \mathcal{A}$ and $\conv G$, where $(G,u)$ is a unital lattice-ordered group with $\mathcal{A}=Γ(G,u)$.

Jakubík, J. 2008. Sequential convergences on MV-algebras without Urysohn's axiom. In Mathematica Slovaca, vol. 58, no.3, pp. 289-300. 0139-9918.

Jakubík, J. (2008). Sequential convergences on MV-algebras without Urysohn's axiom. Mathematica Slovaca, 58(3), 289-300. 0139-9918.