Iséki algebras. Connection with BL algebras

ABSTRACT. State MV-algebras were introduced by Flaminio and Montagna as MV-algebras with internal states. Di Nola and Dvurečenskij presented the notion of state-morphism MV-algebra which is a stronger variation of a state MV-algebra. Rachůnek andŠalounová introduced state GMV-algebras (pseudo-MV algebras) and state-morphism GMV-algebras, while the state BL-algebras and state-morphism BL-algebras were defined by Ciungu, Dvurečenskij and Hyčko. Recently, Dvurečenskij, Rachůnek andŠalounová presented state R -monoids and state-morphism R -monoids. In this paper we study these concepts for more general fuzzy structures, namely pseudo-hoops and we present state pseudo-hoops and state-morphism pseudo-hoops.

“…It follows that all the properties of a pseudo-BCK algebra with pseudo-product proved in [40] and [41] are also valid in a pseudo-hoop.…”

Section: ò ø óò 21º ([38]) a Pseudo-hoop Is An Algebra (Amentioning

confidence: 78%

Bounded pseudo-hoops with internal states

Ciungu

2013

“…We conclude with a brief discussion of a class of BCK-monoids, which, at first sight, appears to generalize the notion of a left pseudo Iséki algebra of Iorgulescu [15].…”

Section: N V Subrahmanyammentioning

confidence: 99%

“…Sections 4 and 5 deal with hoop monoids and Wajsberg monoids (which correspond respectively to GBL-algebras and GMV-algebras of Galatos and Tsinakis [9]) and their special properties. Finally, in Section 6, we study Iséki monoids, which generalize the concept of a left pseudo-Iséki algebra of Iorgulescu [15].…”

Section: Bck-monoid; Then C Is Isomorphic To the Direct Product Of Anmentioning

confidence: 99%

BCK-monoids

Subrahmanyam

2010

ABSTRACT. We generalize the notions of a pseudo BCK-algebra and a residuated lattice by introducing, respectively, extended BCK-algebras and BCK-monoids, and prove a decomposition theorem for BCK-monoids, generalising a decomposition theorem of Galatos and Tsinakis for GBL-algebras. Also, we specialise this theorem to hoop monoids and Wajsberg monoids, which generalize, respectively, GBL-algebras and GMV-algebras. Finally, we include a discussion on Iséki monoids, which extend the concept of left pseudo Iséki algebras of Iorgulescu.

“…As a common algebraic abstract of fuzzy logics (commutative and non-commutative), residuated lattices play an important role in t-norm based fuzzy logic systems (see [12][13][14]27,28]), and applied to rough set theory (see [30]). Recently, non-associative fuzzy logics have been a field of intensive research, so residuated lattice ordered groupoids become a research focus (see [2][3][4][5]29]).…”

Section: Introductionmentioning

confidence: 99%

BCC-algebras and residuated partially-ordered groupoid

Zhang

2013

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.