Automorphisms of the semigroup of invertible matrices with nonnegative elements

Bunina, E. I.; Михалев, А. В.

doi:10.1007/s10958-007-0094-5

Cited by 16 publications

(20 citation statements)

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“…For our purposes S n is realized as n × n matrices of the form δ iσ( j) for σ ∈ S n . (This group decomposition result can been seen in [17].) We define the homomorphism π : Ꮽ → Ꮽ/K n = S n to be the homomorphism obtained as a result of the above semidirect product.…”

Section: Ultradiscretizable Matrix Structurementioning

confidence: 92%

An Ultradiscrete Matrix Version of the Fourth Painlevé Equation

Field¹,

Ormerod²

2007

Adv. Differ Equ.

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This paper is concerned with the matrix generalization of ultradiscrete systems. Specifically, we establish a matrix generalization of the ultradiscrete fourth Painlevé equation (ud-P IV ). Well-defined multicomponent systems that permit ultradiscretization are obtained using an approach that relies on a group defined by constraints imposed by the requirement of a consistent evolution of the systems. The ultradiscrete limit of these systems yields coupled multicomponent ultradiscrete systems that generalize ud-P IV . The dynamics, irreducibility, and integrability of the matrix-valued ultradiscrete systems are studied.

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Section: Ultradiscretizable Matrix Structurementioning

confidence: 92%

An Ultradiscrete Matrix Version of the Fourth Painlevé Equation

Field¹,

Ormerod²

2007

Adv. Differ Equ.

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“…Similarly to the previous lemma we can write a formula DT woM any n (M 1 , M 2 , M ), which is true in G n (R) if and only if 2) ), ξ ∈ R * + (see the beginning of the proof of Lemma 9 of the paper [2]). Similarly , the formula…”

Section: Lemmamentioning

confidence: 97%

“…, n). It follows from Lemma 7 ( [2]) that M = Φ N ′ (S ρ ) for some N ′ ∈ Γ n (R). Let us fix some matrix M satisfying the formula Cycle n (M ) (if we do it, then a matrix N ′ is chosen up to multiiplication by matrices commuting with S ρ ).…”

Section: Lemmamentioning

confidence: 99%

“…Beidar and A.V. Mikhalev ([4]), using some results of model theory (namely, the construction of ultrapower and Keisler-Shelah Isomorphism Theorem) formulated a general approach to the problem of elementary equivalence of various algebraic structures.Taking into account some results of the theory of linear groups over rings, they obtained easy proofs of theorems similar to Maltsev's theorem in rather general situations (for linear groups over prime rings, multiplicative semigroups, lattices of submodules, and so on).In 1998 We use notations and definition from [2]. Now we will recall the most necessary definitions.…”

mentioning

confidence: 99%

“…In 1998 We use notations and definition from [2]. Now we will recall the most necessary definitions.…”

mentioning

confidence: 99%

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Elementary equivalence of the semigroup of invertible matrices with nonnegative elements

Bunina

Михалев

2008

J Math Sci

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Let R be a linearly ordered ring with 1/2, G n (R) (n ≥ 3) be the subsemigroup of GL n (R) consisting of all matrices with nonnegative elements. In [1], there is a description of all automorphisms of the semigroup G n (R) in the case where R is a skewfield and n ≥ 2. In [2], there is a description of all automorphisms of the semigroup G n (R) for the case where R is an arbitrary linearly ordered ring with 1/2 and n ≥ 3. In this paper, we classify semigroups G n (R) up to elementary equivalence.Two models U 1 and U 2 of the same first order language L (e. g. two groups, semigroups, or two rings, semirings) are called elementarily equivalent, if every sentence ϕ of the language L is true in the model U 1 if and only if it is true in the model U 2 .Any two finite models of the same language are elementarily equivalent if and only if they are isomorphic. Any two isomorphic models are elementarily equivalent but for infinite models the converse is not true. For example, the field C of complex numbers and the field Q of algebraic numbers are elementarily equivalent but not isomorphic (since they have different cardinalities).The first results in elementary equivalence of linear groups were obtained by A.I. Maltsev in [3]. He proved that the groups G m (K) and G n (K ′ ) (where G = GL, P GL, SL, P SL, m, n > 2, K and K ′ are fields of characteristic 0) are elementarily equivalent if and only if m = n and the fields K and K ′ are elementarily equivalent. In 1992, C.I. Beidar and A.V. Mikhalev ([4]), using some results of model theory (namely, the construction of ultrapower and Keisler-Shelah Isomorphism Theorem) formulated a general approach to the problem of elementary equivalence of various algebraic structures.Taking into account some results of the theory of linear groups over rings, they obtained easy proofs of theorems similar to Maltsev's theorem in rather general situations (for linear groups over prime rings, multiplicative semigroups, lattices of submodules, and so on).In 1998 We use notations and definition from [2]. Now we will recall the most necessary definitions. Suppose that R is a linearly ordered ring, R + is the set of all positive elements, R + ∪ {0} is the set of all nonnegative elements of the ring R. By G n (R) we denote the subsemigroup of GL n (R) consisting of all matrices with nonnegative elements.The set of all invertible elements of R is denoted by R

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