A non-associative generalization of MV-algebras

ABSTRACT. The aim of the paper is to investigate the relationship between BCC -algebras and residuated partially-ordered groupoids. We prove that an integral residuated partially-ordered groupoid is an integral residuated pomonoid if and only if it is a double BCC -algebra. Moreover, we introduce the notion of weakly integral residuated pomonoid, and give a characterization by the notion of pseudo-BCI algebra. Finally, we give a method to construct a weakly integral residuated pomonoid (pseudo-BCI algebra) from any bounded pseudo-BCK algebra with pseudo product and any group.

“…And, x ≤ y → z ⇐⇒ y ≤ x z by Theorem 3.1 (5). Thus, the law of residuation holds and (G, ⊗, →, , ≤, 1) is a weakly integral residuated pogroupoid.…”

Section: On the Other Hand By Proposition 13(4) We Have Ymentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

BCC-algebras and residuated partially-ordered groupoid

Zhang

2013

“…Table 1. P r o o f. It is proved in [14] that in any NMV -algebra ≤ is a partial order such that x ∨ y = (x → y) → y is an upper bound of x, y and x ∧ y is a lower bound.…”

Section: èöóôó× ø óò 21º a Strong Nmv-algebra Is An Mv-algebra If Anmentioning

confidence: 99%

Strong NMV-algebras, commutative basic algebras and naBL-algebras

Zhang

2013

ABSTRACT. The aim of the paper is to investigate the relationship among NMV-algebras, commutative basic algebras and naBL-algebras (i.e., non-associative BL-algebras). First, we introduce the notion of strong NMV-algebra and prove that(1) a strong NMV-algebra is a residuated l-groupoid (i.e., a bounded integral commutative residuated lattice-ordered groupoid); (2) a residuated l-groupoid is commutative basic algebra if and only if it is a strong NMV-algebra. Secondly, we introduce the notion of NMV-filter and prove that a residuated l-groupoid is a strong NMV-algebra (commutative basic algebra) if and only if its every filter is an NMV-filter. Finally, we introduce the notion of weak naBL-algebra, and show that any strong NMV-algebra (commutative basic algebra) is weak naBL-algebra and give some counterexamples.

“…C h a j d a and J . Kü h r [3]. More precisely, they considered algebras (A, ⊕, ¬, 0) of type (2, 1, 0) satisfying the axioms (MV2)-(MV6), where the axiom (MV1) was substituted by two more axioms…”

Section: Introductionmentioning

confidence: 99%

“…These algebras are called NMV-algebras (non-associative MV-algebras) ( [3]). Clearly, every MV-algebra fulfils the axioms (WA) and (H), but the converse is not true.…”

Section: Introductionmentioning

confidence: 99%

Weak MV-algebras

Halaš

Plojhar

2008

ABSTRACT. In a recent paper [CHAJDA, I.-KÜHR, J.: A non-associative generalization of MV-algebras, Math. Slovaca 57, (2007), 301-312], authors introduced and studied a non-associative generalization of MV-algebras called NMV-algebras. In contrast to MV-algebras, sections (i.e. principal filters) in NMV-algebras which are proper (i.e. are not MV-algebras), do not admit a structure of an NMV-algebra with respect to the operations defined in a natural way. The aim of the paper is to present a new class of algebras generalizing MV-algebras but sharing the above property.