2,251 research outputs found
Shuffle on positive varieties of languages
We show there is a unique maximal positive variety of languages which does not contain the language (ab)*. This variety is the unique maximal positive variety satisfying the two following conditions: it is strictly included in the class of rational languages and is closed under the shuffle operation. It is also the unique maximal proper positive variety closed under length preserving morphims. The ordered monoids of the corresponding variety of ordered monoids are characterized as follows: for every pair (a, b) of mutually inverse elements, and for every element z of the minimal ideal of the submonoid generated by a and b, (abzab)^ω ≤ ab. In particular this variety is decidable
Commutative positive varieties of languages
We study the commutative positive varieties of languages closed under various
operations: shuffle, renaming and product over one-letter alphabets
On generating series of finitely presented operads
Given an operad P with a finite Groebner basis of relations, we study the
generating functions for the dimensions of its graded components P(n). Under
moderate assumptions on the relations we prove that the exponential generating
function for the sequence {dim P(n)} is differential algebraic, and in fact
algebraic if P is a symmetrization of a non-symmetric operad. If, in addition,
the growth of the dimensions of P(n) is bounded by an exponent of n (or a
polynomial of n, in the non-symmetric case) then, moreover, the ordinary
generating function for the above sequence {dim P(n)} is rational. We give a
number of examples of calculations and discuss conjectures about the above
generating functions for more general classes of operads.Comment: Minor changes; references to recent articles by Berele and by Belov,
Bokut, Rowen, and Yu are adde
Small conjunctive varieties of regular languages
The author's modification of Eilenberg theorem relates the so-called conjunctive varieties of regular languages with pseudovarieties of idempotent semirings. Recent results by Pastijn and his co-authors lead to the description of the lattice of all (pseudo)varieties of idempotent semirings with idempotent multiplication. We describe here the corresponding 78 varieties of languages
Gr\"obner methods for representations of combinatorial categories
Given a category C of a combinatorial nature, we study the following
fundamental question: how does the combinatorial behavior of C affect the
algebraic behavior of representations of C? We prove two general results. The
first gives a combinatorial criterion for representations of C to admit a
theory of Gr\"obner bases. From this, we obtain a criterion for noetherianity
of representations. The second gives a combinatorial criterion for a general
"rationality" result for Hilbert series of representations of C. This criterion
connects to the theory of formal languages, and makes essential use of results
on the generating functions of languages, such as the transfer-matrix method
and the Chomsky-Sch\"utzenberger theorem.
Our work is motivated by recent work in the literature on representations of
various specific categories. Our general criteria recover many of the results
on these categories that had been proved by ad hoc means, and often yield
cleaner proofs and stronger statements. For example: we give a new, more
robust, proof that FI-modules (originally introduced by Church-Ellenberg-Farb),
and a family of natural generalizations, are noetherian; we give an easy proof
of a generalization of the Lannes-Schwartz artinian conjecture from the study
of generic representation theory of finite fields; we significantly improve the
theory of -modules, introduced by Snowden in connection to syzygies of
Segre embeddings; and we establish fundamental properties of twisted
commutative algebras in positive characteristic.Comment: 41 pages; v2: Moved old Sections 3.4, 10, 11, 13.2 and connected text
to arxiv:1410.6054v1, Section 13.1 removed and will appear elsewhere; v3:
substantial revision and reorganization of section
On shuffle ideals of general algebras
We extend a word language concept called shuffle ideal to general algebras. For this purpose, we introduce the relation SH and show that there exists a natural connection between this relation and the homeomorphic embedding order on trees. We establish connections between shuffle ideals, monotonically ordered algebras and automata, and piecewise testable tree languages
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