Finite-Dimensional Near-Vector Spaces Over Fields of Prime Order

Howell, Karin-Therese; Meyer, J. H.

doi:10.1080/00927870902855549

Cited by 10 publications

(6 citation statements)

References 2 publications

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“…One checks that G acts without fixed points on M and is a subgroup of order 4 of the general linear group GL (3,4). By straightforward calculation we see that E is a matrix semigroup.…”

Section: Constructions Using Semigroups Of Group Endomorphismsmentioning

confidence: 99%

“…The multiplication * 1 of Theorem 4.5 is similar to the action of near-vector spaces, see [4] for details. We do not follow this line of discussion and possible links to the near-vector space construction here.…”

Section: Constructions Using Semigroups Of Group Endomorphismsmentioning

confidence: 99%

“…Example 4.2. Let (M, +) := (Z 4 3 , +) be the 4-dimensional direct sum of the group (Z 3 , +) written as column vectors. In the following, the matrices and vectors will be over the field Z 3 .…”

Section: Constructions Using Semigroups Of Group Endomorphismsmentioning

confidence: 99%

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Units in Near-Rings

Boykett

Wendt

2016

Communications in Algebra

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We investigate near-ring properties that generalize nearfield properties about units. We study zero symmetric near-rings N with identity with two interrelated properties: the units with zero form an additive subgroup of (N, +); the units act without fixedpoints on (N, +). There are many similarities between these cases, but also many differences. Rings with these properties are fields, near-rings allow more possibilities, which are investigated. Descriptions of constructions are obtained and used to create examples showing the two properties are independent but related. Properties of the additive group as a pgroup are determined and it is shown that proper examples are neither simple nor J 2 -semisimple.2010 Mathematics Subject Classification. 16Y30.

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“…One checks that G acts without fixed points on M and is a subgroup of order 4 of the general linear group GL (3,4). By straightforward calculation we see that E is a matrix semigroup.…”

Section: Constructions Using Semigroups Of Group Endomorphismsmentioning

confidence: 99%

Section: Constructions Using Semigroups Of Group Endomorphismsmentioning

confidence: 99%

See 1 more Smart Citation

Units in Near-Rings

Boykett

Wendt

2016

Communications in Algebra

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“…In [10] van der Walt showed how to construct an arbitrary finite-dimensional near vector space, using a finite number of near-fields, all having isomorphic 'multiplicative' semigroups. In [6] this construction is used to characterize all finite-dimensional near-vector spaces over F p , where p is a prime. These results were extended to all finite dimension near vector spaces over arbitrary finite fields in [7].…”

Section: Introductionmentioning

confidence: 99%

On the Theory of Near-Vector Spaces

Howell

Chistyakov

2018

J Math Sci

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In this paper we study near vector spaces over a commutative F from a model theoretic point of view. In this context we show regular near vector spaces are in fact vector spaces. We find that near vector spaces are not first order axiomatisable, but that finite block near vector spaces are. In the latter case we establish quantifier elimination, and that the theory is controlled by which elements of the pointwise additive closure of F are automorphisms of the near vector space.

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“…In [12] van der Walt showed how to construct an arbitrary finite-dimensional nearvector space, using a finite number of near-fields, all having isomorphic multiplicative semigroups. This construction was used in [7] to characterise all finite dimensional near-vector spaces over Z p , for p a prime. These results were extended in [8] to all finite dimensional near-vector spaces over arbitrary finite fields.…”

Section: Introductionmentioning

confidence: 99%