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.

Over Absolute Valued Algebras with Central Element not Necassary Idempotent

We study the absolute valued algebras containing a central element non necessary idempotent. We determine the absolute valued algebras containing a central element if we add some requirements. Also we gives a classification of finite-dimensional absolute valued algebras containing a generalized left unit and central element.

Absolute Valued Algebras with Strongly One Sided Unit

We classify the absolute valued algebras with strongly left unit of dimension <=. Also we prove that every 8-dimensional absolute valued algebra with strongly left unit contain a 4-dimensional subalgebra, next we determine the form of theirs algebras by the duplication process.

On the Arens product and approximate identity in locally convex algebras

Let A' and A'' be the dual and bidual spaces of a locally convex algebra A with dual and weak* topology, respectively. In this paper, we show that A has a bounded right (left) approximate identity if and only if A'' has a right (left) unit with respect to the first (second) Arens product.

Eight-dimensional real absolute-valued algebras with left unit whose automorphism group is trivial

We classify, by means of the orthogonal group𝒪7(ℝ), all eight-dimensional real absolute-valued algebras with left unit, and we solve the isomorphism problem. We give an example of those algebras which contain no four-dimensional subalgebras and characterise with the use of the automorphism group those algebras which contain one.