Ideals and their complements in commutative semirings

Chajda, Ivan; Länger, Helmut

doi:10.1007/s00500-018-3493-2

Cited by 3 publications

(2 citation statements)

References 4 publications

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“…There exist two different versions of the concept of a semiring, namely semirings having a unit element ( [5]) and those without such an element ( [3,6]). Since the second version is mostly used in applications, we define the basic concept of this paper as follows:…”

Section: Introductionmentioning

confidence: 99%

Directly decomposable ideals and congruence kernels of commutative semirings

Chajda¹,

Eigenthaler²,

Länger³

2020

MMN

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As pointed out in the monographs [5, 6] on semirings, ideals play an important role despite the fact that they need not be congruence kernels as in the case of rings. Hence, having two commutative semirings S 1 and S 2 , one can ask whether an ideal I of their direct product S = S 1 × S 2 can be expressed in the form I 1 × I 2 where I j is an ideal of S j for j = 1, 2. Of course, the converse is elementary, namely if I j is an ideal of S j for j = 1, 2 then I 1 × I 2 is an ideal of S 1 × S 2. Having a congruence Θ on a commutative semiring S, its 0-class is an ideal of S, but not every ideal is of this form. Hence, the lattice Id S of all ideals of S and the lattice Ker S of all congruence kernels (i.e. 0-classes of congruences) of S need not be equal. Furthermore, we show that the mapping Θ → [0]Θ need not be a homomorphism from Con S onto Ker S. Moreover, the question arises when a congruence kernel of the direct product S 1 × S 2 of two commutative semirings can be expressed as a direct product of the corresponding kernels on the factors. In the paper we present necessary and sufficient conditions for such direct decompositions both for ideals and for congruence kernels of commutative semirings. We also provide sufficient conditions for varieties of commutative semirings to have directly decomposable kernels.

show abstract

Section: Introductionmentioning

confidence: 99%

Directly decomposable ideals and congruence kernels of commutative semirings

Chajda¹,

Eigenthaler²,

Länger³

2020

MMN

Self Cite

View full text Add to dashboard Cite

show abstract

“…Semirings play an important role both in algebra and applications. There exist two different versions of this concept, namely semirings having a unit element ( [5]) and those without such an element ( [3] and [6]). Since the second version is mostly used in applications, we define the basic concept of this paper as follows: Definition 1.1.…”

Section: Introductionmentioning

confidence: 99%

Directly decomposable ideals and congruence kernels of commutative semirings

Chajda¹,

Eigenthaler²,

Länger³

2019

Preprint

Self Cite

View full text Add to dashboard Cite

As pointed out in the monographs [5] and [6] on semirings, ideals play an important role despite the fact that they need not be congruence kernels as in the case of rings. Hence, having two commutative semirings S 1 and S 2 , one can ask whether an ideal I of their direct product S = S 1 × S 2 can be expressed in the form I 1 × I 2 where I j is an ideal of S j for j = 1, 2. Of course, the converse is elementary, namely if I j is an ideal of S j for j = 1, 2 then I 1 × I 2 is an ideal of S 1 × S 2 . Having a congruence Θ on a commutative semiring S, its 0-class is an ideal of S, but not every ideal is of this form. Hence, the lattice Id S of all ideals of S and the lattice Ker S of all congruence kernels (i.e. 0-classes of congruences) of S need not be equal. Furthermore, we show that the mapping Θ → [0]Θ need not be a homomorphism from Con S onto Ker S. Moreover, the question arises when a congruence kernel of the direct product S 1 × S 2 of two commutative semirings can be expressed as a direct product of the corresponding kernels on the factors. In the paper we present necessary and sufficient conditions for such direct decompositions both for ideals and for congruence kernels of commutative semirings. We also provide sufficient conditions for varieties of commutative semirings to have directly decomposable kernels.

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Ideals in Bipolar Quantum Linear Algebra

Laipaporn,

Khachorncharoenkul

2024

Symmetry

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Since bipolar quantum linear algebra (BQLA), under two operations–-addition and multiplication—demonstrates the properties of semirings, and since ideals play an important role in abstract algebra, our results are compelling for the ideals of a semiring. In this article, we investigate the characteristics of ideals, principal ideals, prime ideals, maximal ideals, and the smallest ideal containing any nonempty subset. By applying elementary real analysis, particularly the infimum, our main result is stated as follows: for any closed set I in BQLA, I is a nontrivial proper ideal if and only if there exists c∈(0,1] such that I=(−x,y)∈R2|cx≤y≤xcandx,y≥0. This shows that its shape has to be symmetric with the graph y=−x.

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Ideals and their complements in commutative semirings

Cited by 3 publications

References 4 publications

Directly decomposable ideals and congruence kernels of commutative semirings

Directly decomposable ideals and congruence kernels of commutative semirings

Directly decomposable ideals and congruence kernels of commutative semirings

Ideals in Bipolar Quantum Linear Algebra

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