pseudo-BCI Algebras Research Articles

Pseudo-BCI algebras with derivations

AbstractIn this paper we define two types of implicative derivations on pseudo-BCI algebras, we investigate their properties and we give a characterization of regular implicative derivations of type II. We also define the notion of a $d$-invariant deductive system of a pseudo-BCI algebra $A$ proving that $d$ is a regular derivation of type II if and only if every deductive system on $A$ is $d$-invariant. It is proved that a pseudo-BCI algebra is $p$-semisimple if and only if the only regular derivation of type II is the identity map. Another main result consists of proving that the set of all implicative derivations of a $p$-semisimple pseudo-BCI algebra forms a commutative monoid with respect to function composition. Two types of symmetric derivations on pseudo-BCI algebras are also introduced and it is proved that in the case of $p$-semisimple pseudo-BCI algebras the sets of type II implicative derivations and type II symmetric derivations are equal.

On derivation of pseudo BCI-algebras

In this paper, we introduce the notion of derivation of pseudo BCI-algebras and investigate some related properties.

Some properties of pseudo-BCI algebras

Some properties of pseudo-BCI algebras

An injective pseudo-BCI algebra is trivial

An injective pseudo-BCI algebra is trivial

The Structure Theorems of Pseudo-BCI Algebras in Which Every Element is Quasi-Maximal

For mathematical fuzzy logic systems, the study of corresponding algebraic structures plays an important role. Pseudo-BCI algebra is a class of non-classical logic algebras, which is closely related to various non-commutative fuzzy logic systems. The aim of this paper is focus on the structure of a special class of pseudo-BCI algebras in which every element is quasi-maximal (call it QM-pseudo-BCI algebras in this paper). First, the new notions of quasi-maximal element and quasi-left unit element in pseudo-BCK algebras and pseudo-BCI algebras are proposed and some properties are discussed. Second, the following structure theorem of QM-pseudo-BCI algebra is proved: every QM-pseudo-BCI algebra is a KG-union of a quasi-alternating BCK-algebra and an anti-group pseudo-BCI algebra. Third, the new notion of weak associative pseudo-BCI algebra (WA-pseudo-BCI algebra) is introduced and the following result is proved: every WA-pseudo-BCI algebra is a KG-union of a quasi-alternating BCK-algebra and an Abel group.

The lattice generated by hesitant fuzzy filters in pseudo-BCI algebras

The lattice generated by hesitant fuzzy filters in pseudo-BCI algebras

Neutrosophic Hesitant Fuzzy Subalgebras and Filters in Pseudo-BCI Algebras

The notions of the neutrosophic hesitant fuzzy subalgebra and neutrosophic hesitant fuzzy filter in pseudo-BCI algebras are introduced, and some properties and equivalent conditions are investigated. The relationships between neutrosophic hesitant fuzzy subalgebras (filters) and hesitant fuzzy subalgebras (filters) is discussed. Five kinds of special sets are constructed by a neutrosophic hesitant fuzzy set, and the conditions for the two kinds of sets to be filters are given. Moreover, the conditions for two kinds of special neutrosophic hesitant fuzzy sets to be neutrosophic hesitant fuzzy filters are proved.

On pseudo-CI algebras

In this paper, we introduce the notion of pseudo-CI algebras and investigate some of their properties. It is a generalization of the notion of pseudo-BE algebras, pseudo-BCK algebras and pseudo-MV algebras. We give and provide some conditions for a pseudo-CI algebra to be a pseudo-BE algebra. Also, we define the class of singular pseudo-CI algebras and show that any singular pseudo-CI algebra is a pseudo-BCI algebra. The adjoint group of a pseudo-CI algebra is defined and studied. Furthermore, distributive pseudo-CI algebras are discussed and showed that these coincide with distributive pseudo-BE algebras.

Multi-granulation rough filters and rough fuzzy filters in Pseudo-BCI algebras

Multi-granulation rough filters and rough fuzzy filters in Pseudo-BCI algebras

Neutrosophic Hesitant Fuzzy Subalgebras and Filters in Pseudo-BCI Algebras

The notions of the neutrosophic hesitant fuzzy subalgebra and neutrosophic hesitant fuzzy filter in pseudo-BCI algebras are introduced, and some properties and equivalent conditions are investigated. The relationships between neutrosophic hesitant fuzzy subalgebras (filters) and hesitant fuzzy subalgebras (filters) is discussed. Five kinds of special sets are constructed by a neutrosophic hesitant fuzzy set, and the conditions for the two kinds of sets to be filters are given. Moreover, the conditions for two kinds of special neutrosophic hesitant fuzzy sets to be neutrosophic hesitant fuzzy filters are proved.

Monadic pseudo BCI-algebras and corresponding logics

We introduce the notion of monadic pseudo BCI-algebras and study some related properties. Then, we introduce monadic filters and monadic congruences of monadic pseudo BCI-algebras and discuss the relations between them. We proved that there is a one-to-one correspondence between the set of closed m-congruence relations and the set of normal closed m-filters in a monadic pseudo BCI-algebra. Moreover, we introduce a notion of strong residuated mappings and study the relation between monadic operators and strong residuated mappings in pseudo BCI-algebras. Let A be a pseudo BCI-algebra and \(f:A\rightarrow A\) be a mapping, we obtain that \((f, f^+)\) is a monadic operator on A if and only if f is a strong residuated mapping on A where \(f^+\) is the residual of f. Also we exhibit an axiom system of monadic pseudo BCI-logic, which enrich the language of pseudo BCI-logics. Based on the monadic pseudo BCI-algebras, we prove the completeness and soundness of the monadic pseudo BCI-logic propositional system. Finally, using provable formula set, normal subset and monadic subset in the set of all formulas of a monadic pseudo BCI-logic \(\mathcal {L}\), we characterize filters, normal filters and monadic normal filters in a monadic pseudo BCI-algebra, respectively.

Soft set theoretical approach to pseudo-BCI algebras

Soft set theoretical approach to pseudo-BCI algebras

On branchwise commutative pseudo-BCH algebras

Basic properties of branches of pseudo-BCH algebras are described. Next, the concept of a branchwise commutative pseudo-BCH algebra is introduced. Some conditions equivalent to branchwise commutativity are given. It is proved that every branchwise commutative pseudo-BCH algebra is a pseudo-BCI algebra.

Fuzzy anti-grouped filters and fuzzy normal filters in pseudo-BCI algebras

Fuzzy anti-grouped filters and fuzzy normal filters in pseudo-BCI algebras

States on Pseudo-BCI Algebras

In this paper, we discuss the structure of pseudo-BCI algebras and get that any pseudo-BCI algebra is a union of it's branches. We introduce the notion of local bounded pseudo-BCI algebras and study some related properties. Moreover we define two operations $\wedge_1$, $\wedge_2$ in a local bounded pseudo-BCI algebra $A$ and two local operations $\vee_1$ and $\vee_2$ in $V(a)$ for $a\in M(A)$. We show that in a local $\wedge_1$($\wedge_2$)-commutative local bounded pseudo-BCI algebra $A$, $(V(A),\wedge_1,\vee_1)$($(V(A),\wedge_2,\vee_2)$) forms a lattice for all $a\in M(a)$. We define a Bosbach state on a local bounded pseudo-BCI algebra. Then we give two examples of local bounded pseudo-BCI algebras to show that there is local bounded pseudo-BCI algebras having a Bosbach state but there is some one having no Bosbach states. Moreover we discuss some basic properties about Bosbach states. If $s$ is a Bosbach state of a local bounded pseudo-BCI algebra $A$, we prove that $A/ker(s)$ is equivalent to an MV-algebra. We also introduce the notion of state-morphisms on local bounded pseudo-BCI algebras and discuss the relations between Bosbach states and state-morphisms. Finally we give some characterization of Bosbach states.

On some fuzzy filters in pseudo-BCI algebras.

Some new properties of fuzzy associative filters (also known as fuzzy associative pseudo-filters), fuzzy p-filter (also known as fuzzy pseudo-p-filters), and fuzzy a-filter (also known as fuzzy pseudo-a-filters) in pseudo-BCI algebras are investigated. By these properties, the following important results are proved: (1) a fuzzy filter (also known as fuzzy pseudo-filters) of a pseudo-BCI algebra is a fuzzy associative filter if and only if it is a fuzzy a-filter; (2) a filter (also known as pseudo-filter) of a pseudo-BCI algebra is associative if and only if it is an a-filter (also call it pseudo-a filter); (3) a fuzzy filter of a pseudo-BCI algebra is fuzzy a-filter if and only if it is both a fuzzy p-filter and a fuzzy q-filter.

Quantum B-algebras

Abstract The concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic. The logic of quantales and its algebraic semantics manifests itself in a class of partially ordered algebras with a pair of implicational operations recently introduced as quantum B-algebras. Implicational algebras like pseudo-effect algebras, generalized BL- or MV-algebras, partially ordered groups, pseudo-BCK algebras, residuated posets, cone algebras, etc., are quantum B-algebras, and every quantum B-algebra can be recovered from its spectrum which is a quantale. By a two-fold application of the functor “spectrum”, it is shown that quantum B-algebras have a completion which is again a quantale. Every quantale Q is a quantum B-algebra, and its spectrum is a bigger quantale which repairs the deficiency of the inverse residuals of Q. The connected components of a quantum B-algebra are shown to be a group, a fact that applies to normal quantum B-algebras arising in algebraic number theory, as well as to pseudo-BCI algebras and quantum BL-algebras. The logic of quantum B-algebras is shown to be complete.

BCC-algebras and residuated partially-ordered groupoid

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.