###### An Injective Pseudo-BCI Algebra is Trivial

## Abstract

Injective pseudo-BCI algebras are studied. There is shown that the only injective pseudo-BCI algebra is the trivial one.

###### Left Zeroid and Right Zeroid Elements of Γ-Semirings

## Abstract

In this paper we introduce the notion of a left zeroid and a right zeroid of Γ -semirings. We prove that, a left zeroid of a simple Γ-semiring M is regular if and only if M is a regular Γ -semiring.

###### Introducing Fully Up-Semigroups

## Abstract

In this paper, we introduce some new classes of algebras related to UP-algebras and semigroups, called a left UP-semigroup, a right UP-semigroup, a fully UP-semigroup, a left-left UP-semigroup, a right-left UP-semigroup, a left-right UP-semigroup, a right-right UP-semigroup, a fully-left UP-semigroup, a fully-right UP-semigroup, a left-fully UP-semigroup, a right-fully UP-semigroup, a fully-fully UP-semigroup, and find their examples.

###### Kleene Algebra of Partial Predicates

## Summary

We show that the set of all partial predicates over a set *D* together with the disjunction, conjunction, and negation operations, defined in accordance with the truth tables of S.C. Kleene’s strong logic of indeterminacy [17], forms a Kleene algebra. A Kleene algebra is a De Morgan algebra [3] (also called quasi-Boolean algebra) which satisfies the condition *x* ∧*¬:x* ⩽ *y* ∨*¬ :y* (sometimes called the normality axiom). We use the formalization of De Morgan algebras from [8].

The term “Kleene algebra” was introduced by A. Monteiro and D. Brignole in [3]. A similar notion of a “normal i-lattice” had been previously studied by J.A. Kalman [16]. More details about the origin of this notion and its relation to other notions can be found in [24, 4, 1, 2]. It should be noted that there is a different widely known class of algebras, also called Kleene algebras [22, 6], which generalize the algebra of regular expressions, however, the term “Kleene algebra” used in this paper does not refer to them.

Algebras of partial predicates naturally arise in computability theory in the study on partial recursive predicates. They were studied in connection with non-classical logics [17, 5, 18, 32, 29, 30]. A partial predicate also corresponds to the notion of a partial set [26] on a given domain, which represents a (partial) property which for any given element of this domain may hold, not hold, or neither hold nor not hold. The field of all partial sets on a given domain is an algebra with generalized operations of union, intersection, complement, and three constants (0, 1, *n* which is the fixed point of complement) which can be generalized to an equational class of algebras called DMF-algebras (De Morgan algebras with a single fixed point of involution) [25]. In [27] partial sets and DMF-algebras were considered as a basis for unification of set-theoretic and linguistic approaches to probability.

Partial predicates over classes of mathematical models of data were used for formalizing semantics of computer programs in the composition-nominative approach to program formalization [31, 28, 33, 15], for formalizing extensions of the Floyd-Hoare logic [7, 9] which allow reasoning about properties of programs in the case of partial pre- and postconditions [23, 20, 19, 21], for formalizing dynamical models with partial behaviors in the context of the mathematical systems theory [11, 13, 14, 12, 10].

###### Topological Up-Algebras

## Abstract

In this paper, we introduce the notion of topological UP-algebras and several types of subsets of topological UP-algebras, and prove the generalization of these subsets. We also introduce the notions of quotient topological spaces of topological UP-algebras and topological UP-homomorphisms. Furthermore, we study the relation between topological UP-algebras, Hausdor spaces, discrete spaces, and quotient topological spaces, and prove some properties of topological UP-algebras.

###### The Up-Isomorphism Theorems for Up-Algebras

## Abstract

In this paper, we construct the fundamental theorem of UP-homomorphisms in UP-algebras. We also give an application of the theorem to the first, second, third and fourth UP-isomorphism theorems in UP-algebras.

###### (Fuzzy) Isomorphism Theorems of Soft Γ-Hyperrings

## Abstract

Soft set theory, introduced by Molodtsov, has been considered as an effective mathematical tool for modeling uncertainties. In this paper, we apply soft sets to Γ-hyperrings. The concept of soft Γ-hyperrings is first introduced. Then three iso-morphism theorems of soft Γ-hyperrings are established. Finally, we derive three fuzzy isomorphism theorems of soft Γ-hyperrings.

###### Strong ideals and horizontal ideals in pseudo-BCH-algebras

## Abstract

In this paper we define strong ideals and horizontal ideals in pseudo-BCH-algebras and investigate the properties and characterizations of them.

###### Study of Hilbert algebras in point of filters

## Abstract

The aim of this work is to introduce some types of filters in Hilbert algebras. Some theorems are stated and proved which determine the relationship between these notions and other filters of Hilbert algebra and by some examples we show that these concepts are different. The relationships between these filters and quotient algebras that are constructed via these filters are described.

###### Residuated Structures Derived from Commutative Idempotent Semirings

## Abstract

Since the reduct of every residuated lattice is a semiring, we can ask under what condition a semiring can be converted into a residuated lattice. It turns out that this is possible if the semiring in question is commutative, idempotent, G-simple and equipped with an antitone involution. Then the resulting residuated lattice even satisfies the double negation law. Moreover, if the mentioned semiring is finite then it can be converted into a residuated lattice or join-semilattice also without asking an antitone involution on it. To a residuated lattice **L** which does not satisfy the double negation law there can be assigned a so-called augmented semiring. This can be used for reconstruction of the so-called core *C*(**L**) of **L**. Conditions under which *C*(**L**) constitutes a subuniverse of **L** are provided.