Rational semigroup automata

Elaine Render; Mark Kambites

Home/Archive/arXiv

arXiv · 0708.0947

Rational semigroup automata

Elaine Render·Mark Kambites

View original source ↗Read source document ↗Export metadata ↓

Abstract

We show that for any monoid M, the family of languages accepted by M-automata (or equivalently, generated by regular valence grammars over M) is completely determined by that part of M which lies outside the maximal ideal. Hence, every such family arises as the family of languages accepted by N-automata where N is a simple or 0-simple monoid. A consequence is that every such family is either the class of regular languages, contains all the blind one-counter languages, or is the family of languages accepted by G-automata for G a non-locally-finite torsion group. We consider a natural extension of the usual definition which permits the automata to utilise more of the structure of each monoid, and also allows us to define S-automata for S an arbitrary semigroup. In the monoid case, the resulting automata are equivalent to the valence automata with rational target sets} which arise in the theory of regulated rewriting systems. We study the case that the register semigroup is completely simple or completely 0-simple, obtaining a complete characterisation of the classes of languages corresponding to such semigroups in terms of their maximal subgroups. In the process, we obtain a number of results about rational subsets of Rees matrix semigroups which may be of independent interest.

Explore related subjects

math.RA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Elaine Render, Mark Kambites. 2007-08-07. Rational semigroup automata. https://arxiv.org/abs/0708.0947

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

t-Product and t-STP of cubic matrices with an application to hyper-networked systems

Control systems with tensor-valued state transitions require a product that specifies both how coefficients act on each frontal slice and how different slices interact. This paper develops a t-semi-tensor product (t-STP) on cubic matrices that retains the circular coupling of the t-product while allowing rectangular coefficient slices to act on a fixed state space. The construction combines the dimension-keeping semi-tensor product (DK-STP) bridge with circular convolution, overcoming the absence of cross-slice coupling in a slice-wise DK-STP. It provides a compact coefficient description of a structured class of dynamical operators, with fewer stored entries when the coefficient slices have fewer columns than rows. For a fixed number of frontal slices, we establish associative algebra and module structures and describe the associated Lie algebra and Lie groups. These structures make coefficient composition and exponential evolution consistent, while equivalent classical matrix realizations connect the tensor formulation to control analysis of cubic matrix-based dynamics. A specified supply-network game illustrates how the construction organizes interacting chain flows, reproduces the classical trajectories, and supports a globally convergent payoff-gradient adjustment law with explicit damping. The example quantifies coefficient storage while clarifying that the state dimension is unchanged and that the same economy is available to a classical implementation retaining the factorization.

math.RA↗

arXiv2026

Maximal Subsemigroups of Infinite Symmetric Inverse Monoids

The symmetric inverse monoid $I_X$ on a set $X$ consists of all bijective functions whose domain and range are subsets of $X$ under the usual composition and inversion of partial functions. For an arbitrary infinite set $X$, we classify all maximal subsemigroups and maximal inverse subsemigroups of $I_X$ which contain the symmetric group Sym($X$) or any of the following subgroups of Sym($X$): the pointwise stabiliser of a finite subset of $X$, the stabiliser of an ultrafilter on $X$, or the stabiliser of a partition of $X$ into finitely many parts of equal cardinality.

math.RA↗

arXiv2026

Noncommutative resolutions of noncommutative isolated singularities

Noncommutative resolutions of AS-Gorenstein isolated singularities are investigated by Li--Shen--Wu. However, establishing their existence and constructing such resolutions are generally difficult, even when they exist. In this paper, we study conditions under which a commonly graded AS-regular algebra serves as a noncommutative resolution of an AS-Gorenstein isolated singularity. We investigate projective modules over a noetherian commonly graded AS-regular algebra whose endomorphism rings admit resolutions by the underlying regular algebra. This leads to a more general definition of noncommutative resolutions of balanced Cohen--Macaulay isolated singularities. We show that the existence of such resolutions is equivalent to the existence of cluster tilting modules over balanced CM isolated singularities. The corresponding noncommutative analogue of the Bondal-Orlov conjecture is established in dimensions $2$ and $3$. As an application, we study Hopf actions on commonly graded AS-Gorenstein algebras and investigate noncommutative resolutions of invariant rings. We present three examples of noncommutative resolutions, including one in which the noncommutative isolated singularity is not connected graded.

math.RA↗