Non-commutative Multiple-Valued Logic Algebras

BCK algebras were introduced originally by K. Isèki with a binary operation ∗ modeling the set-theoretical difference and with a constant element 0, that is, a least element. Another motivation is from classical and non-classical prepositional calculi modeling logical implications. Such algebras contain as a special subfamily the family of MV-algebras where some important fuzzy structures can be studied. Pseudo-BCK algebras were introduced by G. Georgescu and A. lorgulescu as algebras with “two differences”, a left- and right-difference, instead of one ∗ and with a constant element 0 as the least element. Nowadays pseudo-BCK algebras are used in a dual form, with two implications, → and ⇝ and with one constant element 1, that is, the greatest element. Thus such pseudo-BCK algebras are in the “negative cone” and are also called “left-ones”. More properties of pseudo-BCK algebras and their connection with other fuzzy structures were established by A. lorgulescu. In this chapter we prove new properties of pseudo-BCK algebras with pseudo-product and pseudo-BCK algebras with pseudo-double negation and we show that every pseudo-BCK algebra can be extended to a good one. Examples of proper pseudo-BCK algebras, good pseudo-BCK algebras and pseudo-BCK lattices are given and the orthogonal elements in a pseudo-BCK algebra are characterized. Finally, we define the maximal and normal deductive systems of a pseudo-BCK algebra with pseudo-product and we study their properties.

In this chapter we present the main notions and results regarding the pseudo-hoops, we prove new properties of these structures, we introduce the notions of join-center and cancellative-center of pseudo-hoops and we define and study algebras on subintervals of pseudo-hoops.

Non-commutative residuated lattices, sometimes called pseudo-residuated lattices, biresiduated lattices or generalized residuated lattices, are the algebraic counterparts of substructural logics, i.e. logics which lack at least one of the three structural rules, namely contraction, weakening and exchange. Particular cases of residuated lattices are the full Lambek algebras (FL-algebras), integral residuated lattices and bounded integral residuated lattices (FL _w -algebras).

In this chapter we investigate the properties of a residuated lattice and the lattice of filters of a residuated lattice, we study the Boolean center of an FL _w -algebra and we define and study the directly indecomposable FL _w -algebras. We prove that any linearly ordered FL _w -algebra is directly indecomposable and any subdirectly irreducible FL _w -algebra is directly indecomposable. Finally, the FL _w -algebras of fractions relative to a meet-closed system is introduced and investigated.

In this chapter we study special classes of non-commutative residuated structures: local, perfect and Archimedean structures. The local bounded pseudo-BCK(pP) algebras are characterized in terms of primary deductive systems, while the perfect pseudo-BCK(pP) algebras are characterized in terms of perfect deductive systems. One of the main results consists of proving that the radical of a bounded pseudo-BCK(pP) algebra is normal. We also prove that any linearly ordered pseudo-BCK(pP) algebra and any locally finite pseudo-BCK(pP) algebra are local. Other results state that any local FL _w -algebra and any locally finite FL _w -algebra are directly indecomposable. The classes of Archimedean and hyperarchimedean FL _w -algebras are introduced and it is proved that any locally finite FL _w -algebra is hyperarchimedean and any hyperarchimedean FL _w -algebra is Archimedean.

In this chapter we will present the notion of state for the case of pseudo-BCK algebras. One of the main results consists of proving that any Bosbach state on a good pseudo-BCK algebra is a Riečan state, but conversely it turns out not to be true. Some conditions are given for a Riečan state on a good pseudo-BCK algebra to be a Bosbach state. In contrast to the case of pseudo-BL algebras, we show that there exist linearly ordered pseudo-BCK algebras having no Bosbach states and that there exist pseudo-BCK algebras having normal deductive systems which are maximal, but having no Bosbach states. Some specific properties of states on FL _w -algebras, pseudo-MTL algebras, bounded Rℓ-monoids and subinterval algebras of pseudo-hoops are proved. A special section is dedicated to the existence of states on the residuated structures, showing that every perfect FL _w -algebra admits at least a Bosbach state and every perfect pseudo-BL algebra has a unique state-morphism. Finally, we introduce the notion of a local state on a perfect pseudo-MTL algebra and we prove that every local state can be extended to a Riečan state.

In this chapter we generalize the measures on BCK-algebras introduced by Dvurečenskij and Pulmannova, to pseudo-BCK algebras that are not necessarily bounded. In particular, we show that if A is a downwards-directed pseudo-BCK algebra and m a measure on it, then the quotient over the kernel of m can be embedded into the negative cone of an Abelian, Archimedean ℓ-group as its subalgebra. This result will enable us to characterize nonzero measure-morphisms as measures whose kernel is a maximal deductive system.

We introduce two kinds of generalized states; namely we define generalized states of type I and II, we study their properties and we prove that every strong type II state is an order-preserving type I state. We prove that any perfect FL _w -algebra admits a strong type I and type II state. Some conditions are given for a generalized state of type I on a linearly ordered bounded Rℓ-monoid to be a state operator.

We introduce the notion of generalized state-morphism and we prove that any generalized state morphism is an order-preserving type I state and, in some particular conditions, an order-preserving type I state is a generalized state-morphism. The notion of a strong perfect FL _w -algebra is introduced and it is proved that any strong perfect FL _w -algebra admits a generalized state-morphism. The notion of generalized Riečan state is also given, and the main results are proved based on the notion of Glivenko property defined for the non-commutative case. The main results consist of proving that any order-preserving type I state is a generalized Riečan state and in some particular conditions the two states coincide. We introduce the notion of a generalized local state on a perfect pseudo-MTL algebra A and we prove that, if A is relative free of zero divisors, then every generalized local state can be extended to a generalized Riečan state. The notions of extension property and Horn-Tarski property are introduced. Finally, we outline how the generalized states give an approach of a theory of probabilistic models for non-commutative fuzzy logics associated to a pseudo t-norm.

In this chapter we study the internal states for the more general fuzzy structures, namely the pseudo-hoops, and we present state pseudo-hoops and state-morphism pseudo-hoops. We define the notions of state operator, strong state operator, state-morphism operator, weak state-morphism operator and we study their properties. We prove that every strong state pseudo-hoop is a state pseudo-hoop and any state operator on an idempotent pseudo-hoop is a weak state-morphism operator. One of the main results of the chapter consists of proving that every perfect pseudo-hoop admits a state operator. Other results refer to the connection of the state operators with states and generalized states on a pseudo-hoop. Some conditions are given for a state operator to be a generalized state and for a generalized state to be a state operator.