Implication in MV-algebras

Abstract

We set up an axiomatic system for the logical connective implication within the framework of MV -algebras which generalizes implication in classical logic described similarly by J. C. Abbott. The induced structure (weak implication algebra) turns to be a join-semilattice whose principal filters are MV -algebras.

A non-associative generalization of MV-algebras

“…There exist several equivalent counterparts of MV-algebras; for instance, MV-algebras are term equivalent to bounded weak implication algebras which were introduced in [4] as a generalization of J. C. Abbott's implication algebras (see [1]). We recall that an implication algebra is an algebra (A, →) satisfying the equations…”

Join-semilattices whose sections are residuated PO-monoids

“…The algebras for the implication fragment of the Lukasiewicz logic are LBCK-algebras, i.e., commutative BCK-algebras satisfying prelinearity, which form join-semilattices whose sections are MV-algebras. Actually, semilattices with the property that every section is an MV-algebra lead to commutative BCK-algebras (weak implication algebras [5]) that need not be embedable into an MV-algebra.…”

On implication in MV-algebras

“…

In [1], the authors introduced the notion of a weak implication algebra, which reflects properties of implication in MV-algebras, and demonstrated that the class of weak implication algebras is definitionally equivalent to the class of upper semilattices whose principal filters are compatible MV-algebras. It is easily seen that weak implication algebras are just duals of commutative BCK-algebras.

…”

“…A weak implication algebra is defined in [1] to be an algebra (A, •, 1) of type (2,0) characterised by the axioms…”

“…For brevity, we shall use the term 'BCK*-algebra' for dual BCK-algebras, and 'cBCK*-algebra', for commutative BCK*-algebras.) By Lemma 1(i) of [1], the identity x • x = 1 holds in every weak implication algebra. On the other hand, x • 1 = 1 is one of the standard axioms of BCK*-algebras.…”