Lattice Ordered G􀀀Semirings

Indian Journal Of Science And Technology Pub Date : 2024-03-20 DOI:10.17485/ijst/v17i12.3184

Tilak Raj Sharma, Rajesh Kumar

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Abstract

Objectives: The main objective of this paper is to derive some of the results of lattice ordered semirings, distributive lattice, lattice ideals and morphisms. Methods: To establish the results, we use some conditions like commutativity, simple, multiplicative idempotent, additively idempotent, and finally, use the concept of lattice ideal in semirings. Findings: First we give some examples of lattice ordered semirings and then study some results regarding lattices, distributive lattices, commutative lattice ordered semirings and finally lattice ideals and morphisms. The unique feature of this study is that the concept of gamma is new for the study of lattices. Novelty: We consider a condition (c.f. Theorem 4.1.5) for an additively idempotent semiring due to which it becomes a distributive lattice ordered semiring. Again, in general, the sum of ideals of a semiring need not be ideal. Indeed, and are ideals of is a set of non-negative integers. Clearly, (say) is not a ideal, because , but . However, this condition does not hold in the case of a lattice ordered semiring. AMS Mathematics subject classification (2020): 16Y60. Keywords: Lattices, additive idempotent, multiplicative Γ-idempotent, k-ideal, lattice ideal, Γ-morphism

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网格有序 G􀀀Semirings

研究目的本文的主要目的是推导格子有序语义、分布格子、格子理想和态式的一些结果。方法：为了建立这些结果，我们使用了一些条件，如交换性、简单性、乘法幂等性、加法幂等性，最后，我们还使用了符号中的格理想概念。研究结果首先，我们给出了一些格有序语义的例子，然后研究了有关格、分布格、交换格有序语义的一些结果，最后研究了格理想和态式。本研究的独特之处在于伽马概念是研究格的新概念。新颖性：我们考虑了一个可加可幂半线的条件（参见定理 4.1.5），由于这个条件，它成为了一个可分配的格有序半线。同样，一般来说，一个配系的理想之和不一定是理想的。事实上，和是非负整数集的理想。显然， (说) 不是理想数，因为，但是。然而，这个条件在格有序配系的情况下并不成立。美国数学会数学学科分类（2020）：16Y60.关键词：网格；加法幂等式；乘法Γ幂等式；k理想；网格理想；Γ态

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Indian Journal Of Science And Technology

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