On finite presentations of inverse semigroups with zero having polynomial growth

For any positive integer n > 1 we construct an example of inverse semigroup with n generators and n - 1 defining relations which has cubic growth and at least n generators in any presentation. This semigroup has the same set of identities as the free monogenic inverse semigroup. In particular, we give the first example of a one relation nonmonogenic inverse semigroup having polynomial growth. We also prove that for any positive integer n there exists an inverse semigroup ϒn of deficiency 1 and rank n + 1 such that ϒn has exponential growth and it does not contain nonmonogenic free inverse subsemigroups. Furthermore, ϒn satisfies the identity [[x, y], [z, t]]2 = [[x, y], [z, t]] of quasi-solvability and it contains a free subsemigroup of rank 2.

Download Full-text

GROWTH OF FINITELY PRESENTED REES QUOTIENTS OF FREE INVERSE SEMIGROUPS

International Journal of Algebra and Computation ◽

10.1142/s0218196711006182 ◽

2011 ◽

Vol 21 (01n02) ◽

pp. 315-328 ◽

Cited By ~ 3

Author(s):

L. M. SHNEERSON ◽

D. EASDOWN

Keyword(s):

Inverse Semigroup ◽

Classical Result ◽

Polynomial Growth ◽

Inverse Semigroups ◽

Nilpotent Elements ◽

Free Semigroups ◽

Primitive Words ◽

Rees Quotient ◽

Finitely Presented ◽

Reduced Words

We prove that a finitely presented Rees quotient of a free inverse semigroup has polynomial growth if and only if it has bounded height. This occurs if and only if the set of nonzero reduced words has bounded Shirshov height and all nonzero reduced but not cyclically reduced words are nilpotent. This occurs also if and only if the set of nonzero geodesic words have bounded Shirshov height. We also give a simple sufficient graphical condition for polynomial growth, which is necessary when all zero relators are reduced. As a final application of our results, we give an inverse semigroup analogue of a classical result that characterizes polynomial growth of finitely presented Rees quotients of free semigroups in terms of connections between non-nilpotent elements and primitive words that label loops of the Ufnarovsky graph of the presentation.

Download Full-text

GROWTH AND EXISTENCE OF IDENTITIES IN A CLASS OF FINITELY PRESENTED INVERSE SEMIGROUPS WITH ZERO

International Journal of Algebra and Computation ◽

10.1142/s0218196796000052 ◽

1996 ◽

Vol 06 (01) ◽

pp. 105-121 ◽

Cited By ~ 9

Author(s):

L.M. SHNEERSON ◽

D. EASDOWN

Keyword(s):

Inverse Semigroup ◽

Exponential Growth ◽

Polynomial Growth ◽

Inverse Semigroups ◽

Semigroup Identity ◽

Free Inverse Semigroup ◽

Nontrivial Identity ◽

Rees Quotient ◽

Finitely Presented

We prove that a finitely presented Rees quotient of a free inverse semigroup has polynomial or exponential growth, and that the type of growth is algorithmically recognizable. We prove that such a semigroup has polynomial growth if and only if it satisfies a certain semigroup identity. However we give an example of such a semigroup which has exponential growth and satisfies some nontrivial identity in signature with involution.

Download Full-text

GROWTH OF REES QUOTIENTS OF FREE INVERSE SEMIGROUPS DEFINED BY SMALL NUMBERS OF RELATORS

International Journal of Algebra and Computation ◽

10.1142/s0218196713500082 ◽

2013 ◽

Vol 23 (03) ◽

pp. 521-545 ◽

Cited By ~ 2

Author(s):

D. EASDOWN ◽

L. M. SHNEERSON

Keyword(s):

Inverse Semigroup ◽

Polynomial Growth ◽

Inverse Semigroups ◽

Finite Alphabet ◽

Natural Operator ◽

Generating Set ◽

One Relator Groups ◽

Infinite Set ◽

Rees Quotient ◽

Finitely Presented

We study the asymptotic behavior of a finitely presented Rees quotient S = Inv 〈A|ci = 0(i = 1, …, k)〉 of a free inverse semigroup over a finite alphabet A. It is shown that if the semigroup S has polynomial growth then S is monogenic (with zero) or k ≥ 3. The three relator case is fully characterized, yielding a sequence of two-generated three relator semigroups whose Gelfand–Kirillov dimensions form an infinite set, namely {4, 5, 6, …}. The results are applied to give a best possible lower bound, in terms of the size of the generating set, on the number of relators required to guarantee polynomial growth of a finitely presented Rees quotient, assuming no generator is nilpotent. A natural operator is introduced, from the class of all finitely presented inverse semigroups to the class of finitely presented Rees quotients of free inverse semigroups, and applied to deduce information about inverse semigroup presentations with one or many relations. It follows quickly from Magnus' Freiheitssatz for one relator groups that every inverse semigroup Π = Inv 〈a1, …, an|C = D 〉 has exponential growth if n > 2. It is shown that the growth of Π is also exponential if n = 2 and the Munn trees of both defining words C and D contain more than one edge.

Download Full-text

A basis theorem for free inverse semigroups

Journal of Algebra ◽

10.1016/0021-8693(77)90278-2 ◽

1977 ◽

Vol 49 (1) ◽

pp. 172-190 ◽

Cited By ~ 10

Author(s):

P.R Jones

Keyword(s):

Inverse Semigroups ◽

Basis Theorem

Download Full-text

BSDEs with polynomial growth generators

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953300000216 ◽

2000 ◽

Vol 13 (3) ◽

pp. 207-238 ◽

Cited By ~ 41

Author(s):

Philippe Briand ◽

René Carmona

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Existence And Uniqueness ◽

Polynomial Growth ◽

Probabilistic Representation ◽

Existence And Uniqueness Results ◽

State Variable ◽

Cahn Equation ◽

Uniqueness Results ◽

Terminal Time

In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in the state variable. We deal with the case of a fixed terminal time, as well as the case of random terminal time. The need for this type of extension of the classical existence and uniqueness results comes from the desire to provide a probabilistic representation of the solutions of semilinear partial differential equations in the spirit of a nonlinear Feynman-Kac formula. Indeed, in many applications of interest, the nonlinearity is polynomial, e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.

Download Full-text

A Generalization of Strassen’s Theorem on Preordered Semirings

Order ◽

10.1007/s11083-021-09570-7 ◽

2021 ◽

Author(s):

Péter Vrana

Keyword(s):

Compact Hausdorff Space ◽

Polynomial Growth ◽

Hausdorff Space ◽

Continuous Functions ◽

Equivalent Condition ◽

Real Numbers ◽

Asymptotic Spectrum ◽

Ring Of Continuous Functions ◽

Archimedean Property ◽

Locally Compact Hausdorff Space

AbstractGiven a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.

Download Full-text