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Gelfand residuated lattices

Saeed Rasouli, Amin Dehghani

Abstract

In this paper, a combination of algebraic and topological methods is applied to obtain new and structural results on Gelfand residuated lattices. It is demonstrated that Gelfand's residuated lattices strongly tied up with the hull-kernel topology. Especially, it is shown that a residuated lattice is Gelfand if and only if its prime spectrum, equipped with the hull-kernel topology, is normal. The class of soft residuated lattices is introduced, and it is shown that a residuated lattice is soft if and only if it is Gelfand and semisimple. Gelfand residuated lattices are characterized using the pure part of filters. The relation between pure filters and radicals in a Gelfand residuated lattice is described. It is shown that a residuated lattice is Gelfand if and only if its pure spectrum is homeomorphic to its usual maximal spectrum. The pure filters of a Gelfand residuated lattice are characterized. Finally, it is proved that a residuated lattice is Gelfand if and only if the hull-kernel and the $\mathscr{D}$-topology coincide on the set of maximal filters.

Gelfand residuated lattices

Abstract

In this paper, a combination of algebraic and topological methods is applied to obtain new and structural results on Gelfand residuated lattices. It is demonstrated that Gelfand's residuated lattices strongly tied up with the hull-kernel topology. Especially, it is shown that a residuated lattice is Gelfand if and only if its prime spectrum, equipped with the hull-kernel topology, is normal. The class of soft residuated lattices is introduced, and it is shown that a residuated lattice is soft if and only if it is Gelfand and semisimple. Gelfand residuated lattices are characterized using the pure part of filters. The relation between pure filters and radicals in a Gelfand residuated lattice is described. It is shown that a residuated lattice is Gelfand if and only if its pure spectrum is homeomorphic to its usual maximal spectrum. The pure filters of a Gelfand residuated lattice are characterized. Finally, it is proved that a residuated lattice is Gelfand if and only if the hull-kernel and the -topology coincide on the set of maximal filters.
Paper Structure (4 sections, 42 theorems, 7 equations, 2 figures, 4 tables)

This paper contains 4 sections, 42 theorems, 7 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Residuated lattices
  3. Gelfand residuated lattices
  4. The pure spectrum of a Gelfand residuated lattice

Key Result

Proposition 2.4

Let $\mathfrak{A}$ be a residuated lattice and $F$ be a filter of $\mathfrak{A}$. The following assertions hold, for any $x,y\in A$:

Figures (2)

  • Figure 1: Hasse diagram of $\mathfrak{A}_{6}$
  • Figure 2: Hasse diagram of $\mathfrak{A}_{8}$

Theorems & Definitions (77)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Theorem 2.9
  • ...and 67 more