ON THE PROPERTY OF BEING FROBENIUS OF THE SEMIGROUP ALGEBRA OF A FINITE COMMUTATIVE SEMIGROUP

1968 ◽  
Vol 2 (4) ◽  
pp. 781-797
Author(s):  
I S Ponizovskiĭ
Keyword(s):  
Commutative Semigroup ◽  
Semigroup Algebra ◽  
Finite Commutative Semigroup

scholarly journals On a problem of Baayen and Kruyswijk

1968 ◽  
Vol 16 (2) ◽  
pp. 145-149
Author(s):  
D. A. Burgess
Keyword(s):  
Positive Integer ◽  
Commutative Semigroup ◽  
Finite Semigroup ◽  
Multiplicative Semigroup ◽  
Residue Classes ◽  
Finite Commutative Semigroup

We shall call a finite semigroup S arithmetical if there exists a positive integer N and a monomorphism μ of S into the multiplicative semigroup RN of the ring of residue classes of the integers modulo N. In 1965 P. C. Baayen and D. Kruyswijk [1] posed the problem' Is every finite commutative semigroup arithmetical? ' The purpose of this paper is to answer this question.

The centre of the second dual of a commutative semigroup algebra

1984 ◽  
Vol 95 (1) ◽  
pp. 71-92 ◽  
Author(s):  
D. J. Parsons
Keyword(s):  
Banach Algebra ◽  
Commutative Semigroup ◽  
Topological Semigroup ◽  
Continuous Functions ◽  
Semigroup Algebra ◽  
Measure Algebra ◽  
Semitopological Semigroup ◽  
Borel Measures ◽  
Second Dual ◽  
Right Topological Semigroup

If S is an infinite, discrete, commutative semigroup then the semigroup algebra l1(S) is a commutative Banach algebra. Its dual is l∞(S), which is isometrically iso-morphic to C(βS), the space of continuous functions on the Stone-Čech compactification of S. This fact enables us to identify the second dual of l1(S) with M(βS), the space of bounded regular Borel measures on βS. Endowed with the Arens product the second dual is also a Banach algebra, so it is natural to ask whether a product may be defined in M(βS) without reference to l1(S). In §4 this is shown to be possible even when S is a non-discrete semitopological semigroup, provided that the operation in S may be extended to make βS into a left-topological semigroup in the manner of, for example, [2] where further references may be found. (Note, however, that the construction there is of a right-topological semigroup.) Having done this we may use results on βS to provide information about the measure algebra.

On the radical of semigroup algebras satisfying polynomial identities

1986 ◽  
Vol 99 (1) ◽  
pp. 45-50 ◽  
Author(s):  
Jan Okniński
Keyword(s):  
Commutative Semigroup ◽  
Polynomial Identity ◽  
Jacobson Radical ◽  
Semigroup Algebra ◽  
Group Algebras ◽  
Semigroup Algebras ◽  
Polynomial Identities ◽  
Arbitrary Semigroup

In this paper we will be concerned with the problem of describing the Jacobson radical of the semigroup algebraK[S] of an arbitrary semigroupSover a fieldKin the case where this algebra satisfies a polynomial identity. Recently, Munn characterized the radical of commutative semigroup algebras [9]. The key to his result was to show that, in this situation, the radical must be a nilideal. We are going to extend the latter to the case of PI-semigroup algebras. Further, we characterize the radical by means of the properties ofSor, more precisely, by some groups derived fromS. For this purpose we will exploit earlier results leading towards a characterization of semigroup algebras satisfying polynomial identities [5], [15], which generalized the well known case of group algebras (cf. [13], chap. 5).

Note on the Davenport constant of the multiplicative semigroup of the quotient ring 𝔽p[x] 〈f(x)〉

2016 ◽  
Vol 12 (03) ◽  
pp. 663-669 ◽  
Author(s):  
Haoli Wang ◽  
Lizhen Zhang ◽  
Qinghong Wang ◽  
Yongke Qu
Keyword(s):  
Positive Integer ◽  
Commutative Semigroup ◽  
Polynomial Ring ◽  
Quotient Ring ◽  
Multiplicative Semigroup ◽  
Davenport Constant ◽  
Prime Field ◽  
Group Of Units ◽  
Polynomial Formula ◽  
Finite Commutative Semigroup

Let [Formula: see text] be a finite commutative semigroup. The Davenport constant of [Formula: see text], denoted [Formula: see text], is defined to be the least positive integer [Formula: see text] such that every sequence [Formula: see text] of elements in [Formula: see text] of length at least [Formula: see text] contains a proper subsequence [Formula: see text] with the sum of all terms from [Formula: see text] equaling the sum of all terms from [Formula: see text]. Let [Formula: see text] be a polynomial ring in one variable over the prime field [Formula: see text], and let [Formula: see text]. In this paper, we made a study of the Davenport constant of the multiplicative semigroup of the quotient ring [Formula: see text] and proved that, for any prime [Formula: see text] and any polynomial [Formula: see text] which factors into a product of pairwise non-associate irreducible polynomials, [Formula: see text] where [Formula: see text] denotes the multiplicative semigroup of the quotient ring [Formula: see text] and [Formula: see text] denotes the group of units of the semigroup [Formula: see text].

scholarly journals Lebesgue Decomposition of Non-Commutative Measures

2020 ◽  
Author(s):  
Michael T Jury ◽  
Robert T W Martin
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Fock Space ◽  
Reproducing Kernel Hilbert Space ◽  
Semigroup Algebra ◽  
Space Theory ◽  
Lebesgue Decomposition ◽  
Sesquilinear Forms ◽  
Positive Linear Functionals ◽  
Algebra Theory

Abstract We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $C^{\ast }-$algebra, the $C^{\ast }-$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $C^{\ast }-$algebras; any *−representation of the Cuntz–Toeplitz $C^{\ast }-$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory.

Sign in / Sign up

Export Citation Format

Share Document