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An investigation on the n -fold IVRL-filters in triangle algebras

Saeide Zahiri, Arsham Borumand Saeid (2020)

Mathematica Bohemica

The present study aimed to introduce n -fold interval valued residuated lattice (IVRL for short) filters in triangle algebras. Initially, the notions of n -fold (positive) implicative IVRL-extended filters and n -fold (positive) implicative triangle algebras were defined. Afterwards, several characterizations of the algebras were presented, and the correlations between the n -fold IVRL-extended filters, n -fold (positive) implicative algebras, and the Gödel triangle algebra were discussed.

Annihilators in BCK-algebras

Radomír Halaš (2003)

Czechoslovak Mathematical Journal

We introduce the concepts of an annihilator and a relative annihilator of a given subset of a BCK-algebra 𝒜 . We prove that annihilators of deductive systems of BCK-algebras are again deductive systems and moreover pseudocomplements in the lattice 𝒟 ( A ) of all deductive systems on 𝒜 . Moreover, relative annihilators of C ∈ 𝒟 ( A ) with respect to B ∈ 𝒟 ( A ) are introduced and serve as relative pseudocomplements of C w.r.t. B in 𝒟 ( A ) .

Antiassociative groupoids

Milton Braitt, David Hobby, Donald Silberger (2017)

Mathematica Bohemica

Given a groupoid 〈 G , ☆ 〉 , and k ≥ 3 , we say that G is antiassociative if an only if for all x 1 , x 2 , x 3 ∈ G , ( x 1 ☆ x 2 ) ☆ x 3 and x 1 ☆ ( x 2 ☆ x 3 ) are never equal. Generalizing this, 〈 G , ☆ 〉 is k -antiassociative if and only if for all x 1 , x 2 , ... , x k ∈ G , any two distinct expressions made by putting parentheses in x 1 ☆ x 2 ☆ x 3 ☆ ⋯ ☆ x k are never equal. We prove that for every k ≥ 3 , there exist finite groupoids that are k -antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.

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