The Sheffer stroke operation reducts of basic algebras

Abstract

Abstract:In this study, a term operation Sheffer stroke is presented in a given basic algebra A and the properties of the Sheffer stroke reduct of A are examined. In addition, we qualify such Sheffer stroke basic algebras. Finally, we construct a bridge between Sheffer stroke basic algebras and Boolean algebras.

Sheffer stroke temel cebirlerinden MTL-cebirlerine bir köprü inşası

“…2.9. Lemma [2] Let  = ( B; | ) be a Sheffer stroke basic algebra. A binary relation ≤ is defined on B as follows…”

“…By the time we analyze this reduction, we think about Sheffer stroke operation for algebraic structures. Oner and Senturk introduced a reduction of basic algebras by means of only Sheffer stroke operation which is called Sheffer stroke basic algebra [2]. Sheffer stroke basic algebras play an important role in great numbers of logics as many-valued Łukasiewicz logics, non-classical logics, fuzzy logics and etc.…”

Operations and structures derived from non-associative MV-algebras

“…This is simpler and cheaper than to produce different diods for disjunction, conjunction and negation. Sheffer operations were also introduced in other algebras which form an algebraic semantic of non-classical logics such as orthomodular lattices, ortholattices (Chajda 2005 ) or basic algebras (Oner and Senturk 2017 ). However, all of these algebras have a lattice structure which is not the case for NMV-algebras.…”

“…The second motivation is the fact that the so-called Sheffer operation alias Sheffer stroke was studied by several authors in Boolean algebras (Sheffer 1913), MV-algebras, basic algebras (Oner and Senturk 2017), orthomodular lattices and ortholattices (Chajda 2005), but to our knowledge not on structures which are posets only. In the present paper, we show that we can also introduce and investigate a Sheffer stroke operation in non-associative MV-algebras.…”

Congruences of Sheffer stroke basic algebras

“…Basic algebras were given by Chajda and Emanovský [7], see also Chajda [8] and Chajda et al [9] for more information. In accordance with these studies, Oner and Senturk introduced a reduction of basic algebras by means of only Sheffer stroke operation [18]. Basic algebras are widely used in different non-classical logics because they do not only include orthomodular lattices L = (L; ∨, ∧, ⊥ , 0, 1) where ¬x = x ⊥ and x ⊕ y = (x ∧ y ⊥ ) ∨ y but also yield an axiomatization of the logic of quantum mechanics along with MV-algebras [15], which is obtained an axiomatization of many-valued Lukasiewicz logics; see Chajda [10] and Chajda et al [11].…”