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In the development of the theory of autometrized algebra, various types of research have been conducted. However, there are some properties like convex subalgebra, prime convex subalgebra, meet closed sets, regular convex subalgebra, and convex spectral topology on autometrized algebras that have not been studied yet. In this paper, we define the notions of convex subalgebras and congruence relations on an autometrized algebra. We demonstrate that the collection of all convex subalgebras of an autometrized algebra forms a lattice and distributive. In particular, we will show that there exists a one-to-one correspondence between the set of all convex subalgebras and the set of all congruences on an np-autometrized algebra. Furthermore, we explore prime convex subalgebras, meet closed subsets, and regular convex subalgebras and obtain some related results. For instance, we show that in a semiregular np-autometrized l-algebra, the intersection of a chain of prime convex subalgebra is a prime convex subalgebra. We also prove that any convex subalgebra in an autometrized algebra is the intersection of regular convex subalgebras. Lastly, we introduce the convex spectral topology of proper prime convex subalgebras in an autometrized l-algebra and discuss some fundamental facts. We also prove that a convex spectrum is compact in an np-autometrized l-algebra A if and only if A is generated by some element. Specifically, we demonstrate that the convex spectrum is a T1 - space and Hausdorff space. [ABSTRACT FROM AUTHOR]

In this paper, we investigated some related properties and applied the concept of fuzzy congruence relation to a pseudo-UP algebra X. Additionally, we used their level set to define the fuzzy congruence relation on X. The Cartesian product of two fuzzy congruences and the intersection of two fuzzy congruences are discussed. We also provided the connection between the collection of fuzzy pseudo-UP ideals and the collection of fuzzy congruence relations in pseudo-UP algebras. We then demonstrated that there exists an associated algebra X that is a pseudo-UP algebra for each fuzzy ideal ζ and fuzzy congruence ϕζ generated by ζ. Additionally, we created a congruence relation on a pseudo-UP algebra using the fuzzy pseudo-UP ideal and fuzzy congruence relation ϕζ produced by ζ. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the properties of some elements in autometrized algebra. We also introduce some properties of normal autometrized algebra. Furthermore, we obtain normal autometrized l-algebras among the normal autometrized semialgebras satisfying several conditions and discussing some of its properties. We also prove that a normal autometrized semialgebra satisfying certain conditions is a lattice and distributive. Finally, we establish an equivalent condition between normal autometrized l-algebra and representable autometrized algebra with some conditions. We also prove that in a normal autometrized l-algebra satisfying some conditions, there exists a subset which is a convex normal subalgebra. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of subalgebra and discusses some properties of subalgebra. We also examine different types of ideals and how they behave. We also describe some properties of a homomorphism. Finally, we present quotient algebra and discuss isomorphism theorems. [ABSTRACT FROM AUTHOR]