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Essential Extensions of Filters in Residuated Lattices
Masoud Haveshki and Mahboobeh Mohamadhasani
In this paper, we introduce and analyse the notion of essential extensions of filters in residuated lattices. Using this notion, we characterize residuated lattices which are in a subvariety 𝒱 of the variety ℛℒ of residuated lattices. We use this notion to get a new characterization of some types of filters in residuated lattices. We study the lattice of all essential extensions with respect to a proper ideal 𝐼 of the lattice reduct of 𝐴 and the lattice of all essential extensions of a proper filter 𝐹 of 𝐴 with respect to ideals of the lattice reduct of 𝐴, where 𝐴 is a residuated lattice.
Keywords: Residuated lattice, 𝑀𝑇 𝐿-algebra, 𝐵𝐿-algebra, semi-𝐺-algebra, 𝑀𝑉-algebra, Heyting algebras, Boolean algebra, Stonean residuated lattice, divisible residuated lattice, variety, extended filter, essential extension of filter