Complemented Semiring Satisfying the Identity u + 1 = 1 + u = u
2Complemented Semiring Satisfying the Identity u + 1 = 1 + u = u
Layman Abstract: This paper explores a special type of mathematical structure called a complemented semiring that involves only one variable. We focus on a specific rule within this semiring, which states that adding 1 to any element doesn’t change its value. Using this rule, we prove that the semiring follows certain mathematical properties, making it quasi, weakly, and separative, as well as idempotent (meaning some operations give the same result when repeated).
Additionally, we analyze how addition and multiplication work in this type of semiring. Our findings show that many mathematical results hold true based on the given rule, helping to further understand the structure and behavior of complemented semirings.
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Original Abstract: This paper contains some results on complemented semirings involving one variable. We consider complemented semiring (S1, +, .) satisfying the identity u + 1=1 + u = u for all u, v in S1 then it is proved that (S1, +) is quasi, weakly, separative and also (S1, +, .) is an idempotent. For this complemented semiring we also determine the additive and multiplicative structures. Complemented semiring is holds and equivalent many results by using the identity u + 1 = 1 + u = u for all u, v in S1.
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Please see the book here: https://doi.org/10.9734/bpi/mono/978-93-49238-47-3/CH6