Elementary equivalence of the semigroup of invertible matrices with nonnegative elements

Abstract

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

Extension of Endomorphisms of the Subsemigroup GE 2 + (R) to Endomorphisms of GE 2 + (R[x]), Where R is a Partially-Ordered Commutative Ring Without Zero Divisors

“…In [3], E. I. Bunina and A. V. Mikhalev described all automorphisms of the subsemigroup G n (R), where R is a linearly ordered associative ring with 1/2 and n ≥ 3. In [2], Bunina and Mikhalev found necessary and sufficient conditions for these semigroups to be elementarily equivalent. In [4], E. I. Bunina and P. P. Semenov described the automorphisms of the subsemigroup of the invertible nonnegative matrices of order at least 2 over commutative partially-ordered rings with 1/2.…”

Endomorphisms of the Semigroup G2(r) Over Partially Ordered Commutative Rings Without Zero Divisors and with 1/2

“…In [2], E. I. Bunina and A. V. Mikhalev described all restrictions of automorphisms of G n (R) to GE + n (R) in the case of arbitrary linearly ordered associative rings R with 1/2, n ≥ 3. In [3], E. I. Bunina and A. V. Mikhalev described necessary and sufficient conditions for semigroups considered in [2] to be elementarily equivalent. In [4], E. I. Bunina and P. P. Semenov described automorphisms of G n (R) in the case of commutative partially ordered rings R with 1/2, and in [5] the same authors found necessary and sufficient conditions for semigroups considered above to be elementarily equivalent.…”

Автоморфизмы Полугрупп Обратимых Матриц С Неотрицательными Целыми Элементами

“…В работе [2] Е. И. Бунина и А. В. Михалёв описали все автоморфизмы полугруппы G n (R), если R -произвольное линейно упорядоченное ассоциативное кольцо с 1/2, n 3. В работе [3] Е. И. Бунина и А. В. Михалёв нашли необходимые и достаточные условия для того, чтобы эти полугруппы были элементарно эквивалентны. В работе [4] Е. И. Бунина и П. П. Семёнов описали автоморфизмы полугруппы обратимых неотрицательных матриц порядка > 2 над коммутативными частично упорядоченными кольцами с обратимой двойкой, а в работе [5] нашли необходимые и достаточные условия для их элементарной эквивалентности.…”

“…Мы знаем, что B 1,2 коммутирует со всеми матрицами подстановок, не сдвигающих первые два элемента. Из этого получим, что a 3 3,3 . Отсюда следует, что все слагаемые правой части неравенства (14), не задействованные в выписанных только что неравенствах, должны быть равны нулю.…”