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Benjamin Steinberg

Other affiliations: University of Porto, Bar-Ilan University, Carleton University  ...read more
Bio: Benjamin Steinberg is an academic researcher from City College of New York. The author has contributed to research in topics: Monoid & Semigroup. The author has an hindex of 34, co-authored 266 publications receiving 4016 citations. Previous affiliations of Benjamin Steinberg include University of Porto & Bar-Ilan University.


Papers
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Book
12 May 2009
TL;DR: The q-theory of finite semigroups as mentioned in this paper is a theory that provides a unifying approach to finite semigroup theory via quantization, and it is the only contemporary exposition of the complete theory of the complexity of finite semiigroups.
Abstract: Discoveries in finite semigroups have influenced several mathematical fields, including theoretical computer science, tropical algebra via matrix theory with coefficients in semirings, and other areas of modern algebra. This comprehensive, encyclopedic text will provide the reader - from the graduate student to the researcher/practitioner with a detailed understanding of modern finite semigroup theory, focusing in particular on advanced topics on the cutting edge of research. Key features: (1) Develops q-theory, a new theory that provides a unifying approach to finite semigroup theory via quantization; (2) Contains the only contemporary exposition of the complete theory of the complexity of finite semigroups; (3) Introduces spectral theory into finite semigroup theory; (4) Develops the theory of profinite semigroups from first principles, making connections with spectra of Boolean algebras of regular languages; (5) Presents over 70 research problems, most new, and hundreds of exercises. Additional features: (1) For newcomers, an appendix on elementary finite semigroup theory; (2) Extensive bibliography and index. The q-theory of Finite Semigroups presents important techniques and results, many for the first time in book form, and thereby updates and modernizes the literature of semigroup theory.

325 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the semigroup algebra KS can be described as a convolution algebra of functions on the universal etale groupoid associated to S by Paterson, which is a simultaneous generalization of the author's earlier work on finite inverse semigroups and Paterson's theorem for the universal C ∗ -algebra.

208 citations

Journal ArticleDOI
TL;DR: Solomon's approach to the semigroup algebra of a finite semilattice is extended via Mobius functions to arbitrary finite inverse semigroups, which allows us to explicitly calculate the orthogonal central idempotents decomposing an inverse semigroupgebra into a direct product of matrix algebras over group rings.

120 citations

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TL;DR: The character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using Rota's theory of Mobius inversion were generalized in this paper.

109 citations

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TL;DR: In this paper, the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field is characterized, and applications are given to obtain many new results, as well as easier proofs of several results in the literature.
Abstract: In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier proofs of several results in the literature, involving: triangularizability of finite semigroups; which semigroups have (split) basic semigroup algebras, two-sided semidirect product decompositions of finite monoids; unambiguous products of rational languages; products of rational languages with counter; and Cerny's conjecture for an important class of automata.

103 citations


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08 Dec 2001-BMJ
TL;DR: There is, I think, something ethereal about i —the square root of minus one, which seems an odd beast at that time—an intruder hovering on the edge of reality.
Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

33,785 citations

Book
16 Dec 2017

1,681 citations

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TL;DR: In this paper, a text on rings, fields and algebras is intended for graduate students in mathematics, aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation.
Abstract: This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

1,479 citations

Journal Article
TL;DR: In this article, the fundamental isomorphism theorem of π-algebras is proved and some algebraic properties of Hopf π algebbras are studied.
Abstract: This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

1,322 citations