Semigroups satisfying variable identities

The purpose of this note is to generalize a theorem of Tamura’s [3] providing a self-contained and, we think, more elementary proof than Tamura’s in that it avoids using the theory of contents. Tamura’s result states that a semigroup S satisfies an identify xy=f(x,y) with f(x,y) a word of length greater than 2 which starts with y and ends in x if and only if S is an inflation of a semilattice of groups satisfying the same identity. We investigate semigroups as in Tamura’s Theorem, except that we permit f(x,y) to vary with x and y.     

A problem on semigroups satisfying permutation identities

Let σ be a nontrivial permutation of ordern. A semigroupS is said to be σ-permutable ifx ₁ x ₂ ...x _n =x _σ(1) x _σ(2) ...x _σ(n) , for every (x ₁ ,x ₂,...,x _n )∈S ⁿ . A semigroupS is called(r,t)-commutative, wherer,t are in ℕ^*, ifx ₁ ...x _r x _r+1 ...x _r+t =x _r+1 ...x _r+t x ₁ ...x _r , for every (x ₁ ,x ₂,...,x _r+t ∈S ^r+t . According to a result of M. Putcha and A. Yaqub ([11]), if σ is a fixed-point-free permutation andS is a σ-permutable semigroup then there existsk ∈ ℕ^* such thatS is (1,k)-commutative. In [8], S. Lajos raises up the problem to determine the leastk=k(n) ∈ ℕ^* such that, for every fixed-point-free permutation σ of ordern, every σ-permutable semigroup is also (1,k)-commutative. In this paper this problem is solved for anyn less than or equal to eight and also whenn is any odd integer. For doing this we establish that if a semigroup satisfies a permutation identity of ordern then inevitably it also satisfies some permutation identities of ordern+1.     

Centralizers satisfying polynomial identities

The following results are proved: IfR is a simple ring with unit, and for someaεR witha ⁿ in the center ofR, anyn, such that the centralizer ofa inR satisfies a polynomial identity of degreem, thenR satisfies the standard identity of degreenm. WhenR is not simple,R will satisfy a power of the same standard identity, provided thata andn are invertible inR. These theorems are then applied to show that ifG is a finite solvable group of automorphisms of a ringR, and the fixed points ofG inR satisfy a polynomial identity, thenR satisfies a polynomial identity, providedR has characteristic 0 or characteristicp wherep✗|G|. This research was supported in part by NSF Grant No. GP 29119X.     

On generalized derivations satisfying certain identities

Let R be a prime ring with char R ≠ 2 and let d be a generalized derivation on R. We study the generalized derivation d satisfying any of the following identities:

First-order rigidity of rings satisfying polynomial identities

We prove that every finitely generated Noetherian ring which satisfies a polynomial identity is first-order rigid. This generalizes a result of Aschenbrenner, Khélif, Naziazeno and Scanlon on commutative rings.     

Lie algebras satisfying identities of degree 5

Let be an associative ring with unity, containing 1/6.We prove that every prime Lie -algebra satisfying the identity [(yx)(zx)]x = 0is embedded as a subring of a special form in a three-dimensional simple Lie algebra over some field A. It follows that there exists no central simple Lie algebra which is not three-dimensional and the cube of every inner derivation in which is a derivation. It is proved that if a semiprime Lie algebra over a field satisfies an arbitrary identity of degree 5 (not following from the anticommutativity and Jacobi identities), then it also satisfies the standard identity of degree 5. Essentially used in the proof is the notion of antiderivation. In passing we show that every prime Lie algebra having a nonzero antiderivation satisfies the standard identity of degree 5. Translated fromAlgebra i Logika, Vol. 34, No. 6, pp. 681-705, November-December, 1995.Supported by RFFR grant No. 94-01-00381-a.     

Semigroups satisfying (xy)m = xmym

In this paper we characterize Archimedean semigroups with idempotents satisfying (xy)^m = x^my^m as exactly those semigroups which are a retract extension of a completely simple semigroup satisfying (xy)^m = x^my^m by a nil semigroup satisfying (xy)^m = x^my^m. Regular semigroups satisfying (xy)² = x²y² are exactly those semigroups which are a spined product of a band and a semigroup which is a semilattice of Abelian groups. A semigroup which is a nil extension of a regular semigroup satisfies (xy)² = x²y² if and only if it is a retract extension of a regular semigroup satisfying (xy)² = x²y² by a nil semigroup satisfying (xy)² = x²y²     

Abundant semigroups whose idempotents satisfy permutation identities

The aim of this paper is to study abundant semigroups whose idempotents satisfy permutation identities. After some properties are obtained, the quasi-spined product structure of such semigroups is established while some special cases are investigated. In particular, the structure ofPI-abundant semigroups is obtained.     

Semigroups presented by one relation and satisfying the Church-Rosser property

This paper is devoted to the study of semigroups presented by a single defining relationA=B and satisfying the Church-Rosser property. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 114–118, January, 1997. Translated by A. I. Shtern     

Permanent analogues of determinantal identities related to permutation fixed-points

We present permanent analogues of a determinantal identity due to A. Cayley and a formula computing the determinant of so-called zero-axial matrices, for both the generic commuting and noncommuting cases. The Cayley theorem and its permanental versions are derived using combinatorial interpretation of a classical binomial identity The Theorems concerning zero-axial matrices are gotten by the principle of inclusion-exclusion.     

Asymptotic behaviour of codimensions of p. i. algebras satisfying Capelli identities

Let be a p. i. algebra with 1 in characteristic zero, satisfying a Capelli identity. Then the cocharacter sequence is asymptotic to a function of the form , where and .

     

A new determination of the permutation identities which ensure that a semigroup variety is finitely based

It is shown that every semigroup variety admitting the permutation identity x₁x₂ … x_n = x_1πx_2π … x_nπ is finitely based if and only if 1π ≠ 1 or nπ ≠ n.     

Generalized Bicyclic Semigroups and Jones Semigroups

In this paper, we consider the generalized bicyclic semigroups B_n = a, b | aⁿb = 1 and the Jones semigroups A_n = a, b | aⁿ⁺¹b = a. They are the generalizations of the bicyclic semigroup B = a, b | ab = 1 and its analogous semigroup A = a, b | a²b = a discovered by P.R., Jones in 1987. The word problem for these kinds of semigroups is solved. It is proved that, for n 2, B_n are bisimple right inverse but not inverse semigroups and that the semigroup C = a, b | a²b = a, ab² = b is the smallest idempotent-free homomorphic image of A_n. Moreover, we also prove that A_n and A_m are mutually embeddable but not isomorphic with each other if n m. As a consequence, different kind of -nontrivial [0-]simple semigroups without idempotents are discussed.AMS 1991 Subject Classification: primary 20M10 secondary 20M05.Supported by NNSF of China (19671063) and KSRF of Sichuan Education Committee ([1999]127).