T-Extending Semimodule over Semiring

Alhashemi, Samah; Alhossaini, Asaad M. A.

doi:10.1109/iccitm53167.2021.9677868

Cited by 1 publication

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“…Then (iv) Follows from (iii). Definition 3.27: [9] A subsemimodule 𝒦 of semimodule 𝒟 is called closed if 𝒦 has no proper essential extension in 𝒟. Definition 3.28: [9]A subsemimodule 𝒦 of a semimodule 𝒟 is said to be a closure of a subsemimodule 𝒩 in 𝒟, if 𝒦 is closed and 𝒩 essential in 𝒦.…”

Section: Proofmentioning

confidence: 99%

“…Definition 3.27: [9] A subsemimodule 𝒦 of semimodule 𝒟 is called closed if 𝒦 has no proper essential extension in 𝒟. Definition 3.28: [9]A subsemimodule 𝒦 of a semimodule 𝒟 is said to be a closure of a subsemimodule 𝒩 in 𝒟, if 𝒦 is closed and 𝒩 essential in 𝒦. Proposition 3.29.Let 𝒟 be a distributive semimodule, and ℋ ∈ 𝐿(𝒟), then i. ℋ has a unique complement ℬ in 𝒟, and ℬ coincides with the sum of all subsemimodules of 𝒟 which have zero intersection with ℋ.…”

Section: Proofmentioning

confidence: 99%

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On distributive semimodules

Abdulkhaliq,

Alhossaini

2023

Jour. Kufa Math. Comp.

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This work considers the construction of the concept of distributive property for semimodules. Some characterizations of this property, with some examples are given. Some conditions on semiring or semimodules (like subtractive, semisubtractive, cancellative, and k-cyclic) are required to obtain interesting results. The main results are: Any subsemimodule and factor semimodule of a distributive semimodule is distributive. Moreover, weakly distributive, is also, introduced and investigated. It is found that the semimodule is distributive if and only if each subsemimodule is weak distributive. Taking advantage of the supplemented concept to find some properties of distributive semimodule. Finally, the summand sum and summand intersection properties for distributive semimodules with some conditions are valid.

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Section: Proofmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%