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Mathematical *Logic* Quarterly c ○ WILEY-VCH Verlag Berlin GmbH 2002 Algebraization of the Three-valued *BCK-logic*

"... Abstract. In this paper a definition of n-valued system in the context of the algebraizable logics is proposed. We define and study the variety V3, showing that it is definitionally equivalent to the equivalent quasivariety semantics for the “Three-valued BCK-logic”. As a consequence we find an axio ..."

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Abstract. In this paper a definition of n-valued system in the context of the algebraizable

*logics*is proposed. We define and study the variety V3, showing that it is definitionally equivalent to the equivalent quasivariety semantics for the “Three-valued*BCK-logic*”. As a consequence we find###
*BCK*-algebras Abbreviation: *BCK*

, 1984

"... edit math.chapman.edu/structures 1 Definition 1. A BCK-algebra is a structure A = 〈A, ·, 0 〉 of type 〈2, 0 〉 such that (1): ((x · y) · (x · z)) · (z · y) = 0 (2): x · 0 = x (3): 0 · x = 0 (4): x · y = y · x = 0 = ⇒ x = y Remark: x ≤ y ⇐ ⇒ x · y = 0 is a partial order, with 0 as least element. BCK ..."

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.

*BCK*-algebras provide algebraic semantics for*BCK-logic*, named after the combinators B, C, and K by C. A. Meredith, see [1]. Definition 2. A*BCK*-algebra is a BCI-algebra A = 〈A, ·, 0 〉 such that x · 0 = x Morphisms. Let A and B be*BCK*-algebras. A morphism from A to B is a function h: A → B that is a###
Boolean Skeleton and Pierce representation of Bounded *BCK*-algebras

"... BCK-algebras were introduced by K. Iseki in [6] in order to give an algebraic framework for Meredith’s implicational logic BCK (”BCK logic”). Bounded BCK-algebras were also introduced by Iseki in [7] as BCK-algebras with an additional constant which is interpreted as the lower bound. In fact, they a ..."

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*BCK*-algebras were introduced by K. Iseki in [6] in order to give an algebraic framework for Meredith’s implicational

*logic*

*BCK*(”

*BCK*

*logic*”). Bounded

*BCK*-algebras were also introduced by Iseki in [7] as

*BCK*-algebras with an additional constant which is interpreted as the lower bound. In fact

###
The Relevance Graph of a *BCK*-Formula

"... It is known that the set of BCK-formulas which is provable by the detachment rule of Meredith is identical to the set pts(BCK) of principal type-schemes of BCK-A-terms This paper shows a characterization of the set pts(BCK-f)) of principal type-schemes of BCK-A-terms in ^-normal form To characterize ..."

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variable in 7 Keywords- Typed lambda-calculus, principal types, condensed detachment,

*BCK-logic*1###
Negation and *BCK*-algebras

, 2003

"... Key words BCK-algebras with negation, negation in algebraic structures. ..."

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Tomasz Kowalski THE BOTTOM OF THE LATTICE OF *BCK*-VARIETIES

"... The class of BCK-algebras, introduced in Imai & Iséki [1] as an algebraic counterpart of BCK-logic and extensively studied ever since, can be viewed (dually) as the class of all algebras A = 〈A; ˙−, 0 〉 of the type 〈2, 0 〉 such that A satisfies the following identities: (1) ((x ˙−y) ˙−(x ˙−z)) ˙ ..."

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The class of

*BCK*-algebras, introduced in Imai & Iséki [1] as an algebraic counterpart of*BCK-logic*and extensively studied ever since, can be viewed (dually) as the class of all algebras A = 〈A; ˙−, 0 〉 of the type 〈2, 0 〉 such that A satisfies the following identities: (1) ((x ˙−y) ˙−(x ˙−z###
Biideals in *BCK*/BCI-Bialgebras

"... Abstract. The biideal structure in BCK/BCI-bialgebras is discussed. Relationships be-tween sub-bialgebras, biideals and IC-ideals (and/or CI-ideals) are considered. Conditions for a biideal to be a sub-bialgebra are provided, and conditions for a subset to be a biideal (resp. IC-ideal, CI-ideal) are ..."

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Abstract. The biideal structure in

*BCK*/BCI-bialgebras is discussed. Relationships be-tween sub-bialgebras, biideals and IC-ideals (and/or CI-ideals) are considered. Conditions for a biideal to be a sub-bialgebra are provided, and conditions for a subset to be a biideal (resp. IC-ideal, CI###
ON HEYTING ALGEBRAS AND DUAL *BCK*-ALGEBRAS

"... Abstract. A Heyting algebra is a distributive lattice with im-plication and a dual BCK-algebra is an algebraic system having as models logical systems equipped with implication. The aim of this paper is to investigate the relation of Heyting algebras be-tween dual BCK-algebras. We define notions of ..."

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Abstract. A Heyting algebra is a distributive lattice with im-plication and a dual

*BCK*-algebra is an algebraic system having as models*logical*systems equipped with implication. The aim of this paper is to investigate the relation of Heyting algebras be-tween dual*BCK*-algebras. We define notions