Über Halbringe und Halbkörper. I

Weinert, Hanns Joachim

doi:10.1007/bf02020799

Cited by 34 publications

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“…Semirings are a very natural generalization of rings that has been widely studied, not only from purely algebraic perspective, but also for their applications in cryptography, dequantization, tropical mathematics, non-commutative geometry, and the connection to logic via MV-algebras and latticeordered groups [2], [3], [4], [5], [6], [15], [16], [17], [18], [19], [20], and [21]. We refer the reader to the aforementioned works for further history and references.…”

Section: Theorem 11 ([1] Proposition 12)mentioning

confidence: 99%

Semifields and a theorem of Abhyankar

Kala¹

2017

Commentationes Mathematicae Universitatis Carolinae

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Abstract. Abhyankar [1] proved that every field of finite transcendence degree over Q or over a finite field is a homomorphic image of a subring of the ring of polynomials Z[T 1 , . . . , Tn] (for some n depending on the field). We conjecture that his result can not be substantially strengthened and show that our conjecture implies a well-known conjecture on the additive idempotence of semifields that are finitely generated as semirings [2], [7].

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Section: Theorem 11 ([1] Proposition 12)mentioning

confidence: 99%

Semifields and a theorem of Abhyankar

Kala¹

2017

Commentationes Mathematicae Universitatis Carolinae

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“…This is trivial for (e,&) + (b,,3) = (e,E) with e > b and (a, a) + (e, E) = (e, E) with u<e,andfor(e,cr)+(e,~)=(e,o++)withcr+/?=~ we get (e, a)+(e, P)(e, X) E Z f rom o+pX E L. Now the statement about (BoK)/, follows from group theoretical considerations and (the concept of a lattice ordered group (A, 5, . ), briefly an l-group, we refer to [2], V. Our considerations are based on the following correspondence between lgroups and additively commutative and idempotent semifields (cf [6]. Satz 1kernels corresponding to these morphisms we state (cf.…”

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confidence: 99%

On the structure of semifields and lattice-ordered groups

Weinert

Wiegandt²

1996

Period Math Hung

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Dedicated to the 70-th birthday of Professor Ldszld Fuchs IntroductionIn the present note we continue the study of the radical theory of semifields started in [8]. Th e main purpose of radical theoretical investigations is to prove structure theorems for semisimple objects. Here we apply the general radical theoretical framework developed in [8], and derive structure theorems for semifields, and in particular for lattice ordered groups. We shall prove subdirect decomposition theorems for semifields belonging to the semisimple class of certain radicals, and in the special case of lattice ordered groups we introduce a concrete radical yielding also a direct decomposition theorem for the semisimple complete lattice ordered groups with descending chain condition on kernels. After the preliminaries, in Section 2 we prove sufficient conditions for a class of semifields such that its subdirect closure should be a simple class, and in this way we get a subdirect decomposition for each semifield in the semisimple class considered. Section 3 is devoted to the study of properties and behaviour of simple and subdirectly irreducible semifields. Additively commutative and idempotent semifields can be viewed also as lattice-ordered groups (briefly: I-groups), and so the radical theory of semifields can be applied tc I-groups. This will be done in Section 4. More explicit results can be achieved for complete l-groups. In Section 5 we introduce the anti-real radical, and using specific properties of the I-group of reals, we prove that the&complete l-groups with trivie.

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“…[1], [2]) and some generalizations of this concept will be clarified by equivalent characterizations. [1], [2]) and some generalizations of this concept will be clarified by equivalent characterizations.…”

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confidence: 99%

“…[2]). [2]): If a E S is (left) cancellable in (S, . But even corresponding one-sided or mixed versions of this (replacing 13(q91 by !3, 91, or 13U 91) will prove correct, and we start with left and two-sided cancellability, treated simultaneously.…”

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Multiplicative cancellativity of semirings and semigroups

Weinert

1980

Acta Mathematica Academiae Scientiarum Hungaricae

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In this note, the usual ring-like definition of a multiplicatively cancellative semiring S= (S, +, 9 ) (cf. [1], [2]) and some generalizations of this concept will be clarified by equivalent characterizations. As a matter of fact, the intrinsic parts of these considerations are consequences of statements concerning merely the multiplicative semigroup (S, .) of S, regarded as an arbitrary one, and our results on semirings become those on very general algebras (S, +, .). In this context, we give some applications on nearrings.A semiring S=(S, +, .) is defined as an algebra such that (S, +) and (S, .) are any semigroups, connected by ring-like distributivity. A neutral element o of (S, +), if there is one, is called the zero of the semiring S, and in particular an annihilating zero iff, as for rings, {o} is an ideal of (S, 9 ). For simplicity, expressions like "ar or "S\{o}" are used for any semiring S, being of no consequence if S has no zero.We further denote by 13 [9t] the set of all elements of S being left [right] cancellable in (S, .). Clearly, both are subsemigroups of (S, .) if not empty, and for all x, yES we have (1) xyE13 =" yE13; xyE91 =* xE91.In general, very few can be said about the multiplicative behaviour of a zero of a semiring S (of. [2]). So it is notable that, as claimed in [4], the ring-like concept of multiplicative cancellability (in short: a#o=~aE 13 fq 91), applied to any semiring S, forces a zero of S to be either multiplicatively cancellable, too, or else annihilating. But even corresponding one-sided or mixed versions of this (replacing 13(q91 by !3, 91, or 13U 91) will prove correct, and we start with left and two-sided cancellability, treated simultaneously. To avoid trivial rubs, always ISl=>2 is assumed. DEFINITION 1. A semiring S is called multiplieatively (left) caneellative iff each element ar of S is (left) cancellable in (S, .). PROPOSITION 1..4 semiring S is multiplieatively (left) eaneellative iff one of the following conditions holds: ~) S has no zero, and (S, .) is (left) eaneellative. [1) S has a zero, and (S, .) is (left) eancellative. 7) S has an annihilating zero, and (S',,,{o}, .) is a subsemigroup of (S, .), which in (left) eaneellative?i With obvious definitions, we have for each semiring S: (S~{o}, .) is a subsemigroup ~. S has no proper divisors of zero .~ {o} is a completely prime set. But even for semirings with an annihilating zero this does not imply anything concerning cancellability. We note in this context (cf. [2]): If a E S is (left) cancellable in (S, .), a is no (left) zero divisor, but no conversely.

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Über Halbringe und Halbkörper. I

Cited by 34 publications

References 20 publications

Semifields and a theorem of Abhyankar

Semifields and a theorem of Abhyankar

On the structure of semifields and lattice-ordered groups

Multiplicative cancellativity of semirings and semigroups

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