A ug 2 00 6 Elementary equivalence of the semigroup of invertible matrices with nonnegative elements

Let R be a linearly ordered ring with 1/2, Gn(R) (n ≥ 3) be the subsemigroup of GLn(R) consisting of all matrices with nonnegative elements. In [1], there is a description of all automorphisms of the semigroup Gn(R) in the case where R is a skewfield and n ≥ 2. In [2], there is a description of all automorphisms of the semigroup Gn(R) for the case where R is an arbitrary linearly ordered ring with 1/2 and n ≥ 3. In this paper, we classify semigroups Gn(R) up to elementary equivalence. Two models U1 and U2 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 U1 if and only if it is true in the model U2. 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 Gm(K) and Gn(K ) (where G = GL,PGL, SL, PSL, 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 1998–2004, E.I. Bunina continued to study elementary properties of linear groups. In 1998, the results of A.I. Maltsev were generalized to unitary linear groups over fields with involutions ([5]), and then to unitary linear groups over rings an skewfields with involutions ([6]). In 2001–2004 ([7]) similar results were obtained for Chevalley groups over fields. 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 Gn(R) we denote the subsemigroup of GLn(R) consisting of all matrices with nonnegative elements. The set of all invertible elements of R is denoted by R. The set R+ ∩R ∗ is denoted by R +. If T ⊂ R, then Z(T ) denotes the center of T , Z(T ) = Z(T ) ∩R, Z+(T ) = Z(T ) ∩R+, Z ∗ +(T ) = Z(T ) ∩R ∗ +. Let I = In, Γn(R) be the group consisting of all invertible matrices from Gn(R), Σn be the symmetric group of order n, Sσ be the matrix of the permutation σ ∈ Σn (i. e., the matrix (δiσ(j))), Sn = {Sσ|σ ∈ Σn}, diag[d1, . . . , dn] be the diagonal matrix with elements d1, . . . , dn on the diagonal, where d1, . . . , dn ∈ R ∗ +, Dn(R) be the group of all invertible diagonal matrices from Gn(R), D n (R) be the center of Dn(R). By Kn(R) we denote the subsemigroup in Gn(R) consisting of all matrices of the form