Gődel filters in residuated lattices

Abstract

In this paper, in the spirit of [4], we study a new type of filters in residuated lattices : Gődel filters. So, we characterize the filters for which the quotient algebra that is constructed via these filters is a Gődel algebra and we establish the connections between these filters and other types of filters. Using Gődel filters we characterize the residuated lattices which are Gődel algebras. Also, we prove that a residuated lattice is a Gődel algebra (divisible residuated lattice, MTL algebra, BL algebra) if and only if every filter is a Gődel filter (divisible filter, MTL filter, BL filter). Finally, we present some results about injective Gődel algebras showing that complete Boolean algebras are injective objects in the category of Gődel algebras.

Keywords

• residuated lattice
• gődel algebra
• gődel filter
• injective object
• primary 03b50, 06a06, 03g05
• secondary 03g25, 08a72, 06d35, 06e05, 06b20, 03c05, 18c05