Derivations of Equality Algebras

In this paper, we introduced the concept of derivation on equality algebra E by using the notions of inner and outer derivations. Then we investigated some properties of (inner, outer) derivation and we introduced some suitable conditions that they help us to define a derivation on E. We introduced kernel and fixed point sets of derivation on E and prove that under which condition they are filters of E. Finally we prove that the equivalence relations on (E,⇝, 1) coincide with the equivalence relations on E with derivation d.(2010) MSC: 03G25, 06B10, 06B99.

Radical of filters on equality algebra

In this paper, by using the concept of maximal filter of equality algebra, we introduce radical of equality algebra. Then some equivalence definitions of it and some related properties are investigated. Then by using this notion, we introduce the concept of semi-maximal filter and prime-like filter on equality algebras and the relation between them and other filters of equality algebra are investigated. Finally, by using the notion of prime-like filters, we introduce a topology on equality algebra.

Equality Logic

In this paper, we introduce and study a corresponding logic toequality-algebras and obtain some basic properties of this logic. We provethe soundness and completeness of this logic based on equality-algebrasand local deduction theorem. Then we introduce the concept of (prelinear)equality-algebras and investigate some related properties. Also, westudy -deductive systems of equality-algebras. In particular, we provethat every prelinear equality-algebra is a subdirect product of linearly orderedequality-algebras. Finally, we construct prelinear equality logicand prove the soundness and strong completeness of this logic respect toprelinear equality-algebras.

Ideals in bounded equality algebras

In this paper, the concept of ideal in bounded equality algebras is introduced. With respect to this concepts, some related results are given. In particular, we prove that there is an one-to-one corresponding between congruence relation on an involutive equality algebra and the set of ideals on it. Also, we prove the first isomorphism theorem on equality algebras. Moreover, the notions of prime and Boolean ideals in equality algebras are introduced. Finally, we prove that ideal I of involutive prelinear equality algebra E is a Boolean ideal if and only if E/I is a Boolean algebra.

Generalized state maps and states on pseudo equality algebras

In this paper, we attempt to cope with states in a universal algebraic setting, that is, introduce a notion of generalized state map from a pseudo equality algebra X to an arbitrary pseudo equality algebra Y. We give two types of special generalized state maps, namely, generalized states and generalized internal states. Also, we study two types of states, namely, Bosbach states and Riečan states. Finally, we discuss the relations among generalized state maps, states and internal states (or state operators) on pseudo equality algebras. We verify the results that generalized internal states are the generalization of internal states, and generalized states are the generalization of state-morphisms on pseudo equality algebras. Furthermore, we obtain that generalized states are the generalization of Bosbach states and Riečan states on linearly ordered and involutive pseudo equality algebras, respectively. Hence we can come to the conclusion that, in a sense, generalized state maps can be viewed as a possible united framework of the states and the internal states, the state-morphisms and the internal state-morphisms on pseudo equality algebras.

A non representable infinite dimensional quasi-polyadic equality algebra with a representable cylindric reduct

We construct an infinite dimensional quasi-polyadic equality algebra \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathfrak{A}$ \end{document} such that its cylindric reduct is representable, while \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathfrak{A}$ \end{document} itself is not representable.

Equality algebras

As part of an attempt to capture abstractly the most fundamental properties of algebraic reasoning involving equality, we introduce the notion of an equality algebra. It is a universal algebra A endowed with a binary function =iA × A → L, where L is a meet-semilattice with top element 1, called internalised equality, and satisfying, for all x, y ∈ A,1. (x =ix) = 1; and2. (x =iy)f(x) = (x =iy)f(y), where f is any function A → L derived from the Operations on A, the semilattice operations, and = i.We charecterise internalised equalities in terms of finetly many identities, give examples, and show that all are equivalent to internalised equalities defined in terms of congruences on the underlying algebra. In the special case in which A is an Abelian group or ring, the internalised equality is shown to be equivalent to the dual of a norm-like mapping taking values in semilattice.