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The multiplicative order convergence was studied and investigated on Riesz algebras. This paper deals with Riesz algebras and different topologies on them. In this paper, we investigate Riesz algebras on which we define various kinds of continuities. We give the relation between them under certain specific conditions. We show some relations among locally full, locally convex and locally solid Riesz algebras. Also, we introduce the notions of order and topological continuity of algebraic multiplications on topological Riesz algebras. Also, we extend the multiplication to quotient spaces of Riesz algebras. © 2022, University of Nis. All rights reserved.

The statistically multiplicative convergence in Riesz algebras was studied and investigated with respect to the solid topology. In the present paper, the statistical convergence with the multiplication in Riesz algebras is introduced by developing topology-free techniques using the order convergence in vector lattices. Moreover, we give some relations with the other kinds of convergences such as the order statistical convergence, the $mo$-convergence, and the order convergence.

2-s2.0-85101180282 A net (x(alpha))(alpha is an element of A) in an f-algebra E is called multiplicative order convergent to x is an element of E if vertical bar x(alpha )- x vertical bar . u ->(o) 0 for all u is an element of E+. This convergence was introduced and studied on f-algebras with the order convergence. In this paper, we study a variation of this convergence for normed Riesz algebras with respect to the norm convergence. A net (x(alpha))(alpha is an element of A) in a normed Riesz algebra E is said to be multiplicative norm convergent to x is an element of E if parallel to vertical bar x(alpha) - x vertical bar . u parallel to -> 0 for each u is an element of E+. We study this concept and investigate its relationship with the other convergences, and also we introduce the mn-topology on normed Riesz algebras.© 2021, Hacettepe University. All rights reserved.

In this paper, we introduce the statistically multiplicative convergent sequences in locally solid Riesz algebras with respect to the algebra multiplication and the solid topology. We study on this concept and we give the notion of $\mathbb{st_m}$-bounded sequence, and also, we prove some relations between this convergence and the other convergences such as the order convergence and the statistical convergence in topological spaces. Also, we give some results related to semiprime $f$-algebras., 15 Pages

msufbd X ve E Riesz cebirleri ve p:X ?E_+ monoton bir vektör normu olsun. Böylece (X, p, E) üçlüsü kafes normlu Riesz cebiri olarak adlandırılır. Bu çalışmada, Hahn-Banach teoreminin kafes normlu Riesz cebirlerindeki operatörler için genişletilmesini vereceğiz. Fakat bu çalışmadaki genişleme diğer Hahn-Banach teoremlerinden farlı olmaktadır. Ayrıca bu genişlemenin bazı sonuçlarının olduğunu göstermeketeyiz. Let X and E be Riesz algebras and p:X ?E_+ be a monotone vector norm. Then the triple (X, p, E) is called lattice normed Riesz algebra. In this paper, we prove a generalization of the extension of the Hahn-Banach theorem for operators on the lattice normed Riesz algebras, in which the extension of one-step of that is not similar to the other Hahn-Banach theorems. In addition, we give some applications and results. 727916

Aydin, Abdullah/0000-0002-0769-5752 The multiplicative convergence on Riesz algebras introduced and investigated with respect to norm and order convergences. If is a Riesz space E and is a Riesz algebra then the vector norm mu: X -> E+ can be considered. Then (X, mu, E) is called algebraic lattice normed spaces. A net (x(alpha))(alpha is an element of A) in an (X, mu, E) is said to be multiplicative mu-convergent to x is an element of X if mu(x(alpha) - x).u ->(o) 0 holds for all u is an element of E+. In this paper, the general properties of this convergence are studied.

Let X, Y be compact Hausdorff spaces and let E, F be both Banach lattices and Riesz algebras. In this paper, the following main result shall be proved: If F has no zero-divisor and there exists a Riesz algebraic isomorphism Φ : C ( X , E ) → C ( Y , F ) such that Φ ( f ) has no zero if f has none, then X is homeomorphic to Y and E is Riesz algebraically isomorphic to F.

In this paper we introduce a new class of lattice ordered algebra, which will be called a pseudo-almost f-algebra; namely, a lattice ordered algebra A in which a ∧ b = 0 in A implies ab ∧ ba = 0. We present some fundamental properties of pseudo-almost f-algebras and consider their relationships with various type of lattice ordered algebras; mainely, f-algebras, almost f-algebras and d-algebras. 2010 Mathematics Subject Classification: Primary 46A40, 46B42; Secondary 06F25, 13J25.