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Structures of $(m,n)$-seminearring

M. S. L. Liedokto

TL;DR

This work develops the theory of $(m,n)$-seminearrings as a generalization of $(m,n)$-semirings and related ternary systems. It defines the structure via an $m$-ary addition and an $n$-ary multiplication with a $t$-distributive relation, and then builds the theory of substructures, ideals, unity, homomorphisms, and quotient constructions by congruence. Key contributions include formal definitions of $(m,n)$-subseminearrings, $(m,n)$-ideals (with left, right, and full variants), unity and invertibility concepts, and a factor theorem ensuring well-defined quotient seminearrings. The results extend algebraic structure theory, offering a general framework for more complex multi-ary operations and their interactions, with potential applications in abstract algebra and related fields.

Abstract

This article introduces the $m, n)$-seminearring structure, which is a generalization of $(m, n)$-semiring. This research aims to develop theories of $(m, n)$-seminearring. In particular, the concepts of $(m, n)$-seminearring, $(m, n)$-subseminearring, $(m, n)$-ideal, $(m, n)$-seminearring with unity, homomorphism of $(m, n)$-seminearrings, construction of factor $(m, n)$-seminearring of $(m, n)$-seminearring by congruence relation, and some of its exciting properties are given. The method used in this study is to adopt the theory in $(m, n)$-semiring.

Structures of $(m,n)$-seminearring

TL;DR

This work develops the theory of -seminearrings as a generalization of -semirings and related ternary systems. It defines the structure via an -ary addition and an -ary multiplication with a -distributive relation, and then builds the theory of substructures, ideals, unity, homomorphisms, and quotient constructions by congruence. Key contributions include formal definitions of -subseminearrings, -ideals (with left, right, and full variants), unity and invertibility concepts, and a factor theorem ensuring well-defined quotient seminearrings. The results extend algebraic structure theory, offering a general framework for more complex multi-ary operations and their interactions, with potential applications in abstract algebra and related fields.

Abstract

This article introduces the -seminearring structure, which is a generalization of -semiring. This research aims to develop theories of -seminearring. In particular, the concepts of -seminearring, -subseminearring, -ideal, -seminearring with unity, homomorphism of -seminearrings, construction of factor -seminearring of -seminearring by congruence relation, and some of its exciting properties are given. The method used in this study is to adopt the theory in -semiring.
Paper Structure (9 sections, 12 theorems, 25 equations)

This paper contains 9 sections, 12 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. METHODS
  3. RESULTS AND DISCUSSION
  4. $\boldsymbol{(m,n)}$-seminearring
  5. $\boldsymbol{(m,n)}$-subseminearring
  6. $\boldsymbol{(m,n)}$-ideal
  7. $\boldsymbol{(m,n)}$-seminearring with unity
  8. Homomorphism of $\boldsymbol{(m,n)}$-seminearrings
  9. Factor $\boldsymbol{(m,n)}$-seminearring

Key Result

Lemma 3.8

Let $(R,f,g)$ be an $(m,n)$-seminearring, $S\subseteq R$, and $S\neq \emptyset$. Then $(S,f,g)$ is an $(m,n)$-subseminearring of $(R,f,g)$ if and only if the following two conditions are satisfied.

Theorems & Definitions (37)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Example 3.5
  • Definition 3.6
  • Definition 3.7
  • Lemma 3.8
  • proof
  • Theorem 3.9
  • ...and 27 more