Freeness Problem for Matrix Semigroups of Parikh Matrices

The freeness problem over matrix semigroups and bounded languages

A word is a finite sequence of symbols. Parikh matrix of a word is an upper triangular matrix with ones in the main diagonal and nonnegative integers above the main diagonal which are counts of certain scattered subwords in the word. On the other hand, a picture array, which is a rectangular arrangement of symbols, is an extension of the notion of a word to two dimensions. Parikh matrices associated with a picture array have been introduced, and their properties have been studied. Here, we obtain certain algebraic properties of Parikh matrices of binary picture arrays based on the notions of power, fairness, and a restricted shuffle operator extending the corresponding notions studied in the case of words. We also obtain properties of Parikh matrices of arrays formed by certain geometric operations.

We study the vector ambiguity problem and the vector freeness problem in SL(2,Z). Given a finitely generated n×n matrix semigroup S and an n-dimensional vector x, the vector ambiguity problem is to decide whether for every target vector y=Mx, where M∈S, M is unique. We also consider the vector freeness problem which is to show that every matrix M which is transforming x to Mx has a unique factorization with respect to the generator of S. We show that both problems are NP-complete in SL(2,Z), which is the set of 2×2 integer matrices with determinant 1. Moreover, we generalize the vector ambiguity problem and extend to the finite and k-vector ambiguity problems where we consider the degree of vector ambiguity of matrix semigroups.

In this paper, we show that the problem of determining if the identity matrix belongs to a finitely generated semigroup of 2 × 2 matrices from the modular group PSL2(ℤ) and thus the Special Linear group SL2(ℤ) is solvable in NP. From this fact, we can immediately derive that the fundamental problem of whether a given finite set of matrices from SL2(ℤ) or PSL2(ℤ) generates a group or free semigroup is also decidable in NP. The previous algorithm for these problems, shown in 2005 by Choffrut and Karhumaki, was in EXPSPACE mainly due to the translation of matrices into exponentially long words over a binary alphabet {s, r} and further constructions with a large nondeterministic finite state automaton that is built on these words. Our algorithm is based on various new techniques that allow us to operate with compressed word representations of matrices without explicit expansions. When combined with the known NP-hard lower bound, this proves that the membership problem for the identity problem, the group problem and the freeness problem in SL2(ℤ) are NP-complete.

The group of automorphisms of a zero-divisor graph based on rank one upper triangular matrices

Let F q be a finite field with q elements, n(≥2) a positive integer, and T n (q) the semigroup of all n × n upper triangular matrices over F q . The generalized Cayley graph GCay(T n (q)) of T n (q) is a directed graph with vertex set T n (q), in which there is a directed edge from a vertex A to a distinct vertex B if and only if B = XAY for some X, Y ∈ T n (q). The main result of this article proves that a bijective map σ is an automorphism of GCay(T n (q)) if and only if, for any vertex A of GCay(T n (q)), either σ(A) = P A AQ A or σ(A) = P A JA t JQ A , where A t denotes the transpose of A, , and P A and Q A are invertible upper triangular matrices depending on A.

Parikh matrices have been extensively investigated due to their usefulness in studying subword occurrences in words. Due to the dependency of Parikh matrices on the ordering of the alphabet, the notion of strong [Formula: see text]-equivalence was proposed as an order-independent alternative to [Formula: see text]-equivalence in studying words possessing the same Parikh matrix. This paper introduces and studies the notions of strong [Formula: see text] and strong [Formula: see text] transformations in determining when two ternary words are strongly [Formula: see text]-equivalent. The irreducibility of strong [Formula: see text] transformations is then scrutinized, further exemplified by a structural characterization of irreducible strong [Formula: see text] transformations. The common limitation of these transformations in characterizing strong [Formula: see text]-equivalence is then addressed.

Independence of certain quantities indicating subword occurrences

The cyclic groups via the Pascal matrices and the generalized Pascal matrices

Let [Formula: see text] be a finite field with [Formula: see text] elements, [Formula: see text] a positive integer, [Formula: see text] the semigroup of all [Formula: see text] upper triangular matrices over [Formula: see text] under matrix multiplication, [Formula: see text] the group of all invertible matrices in [Formula: see text], [Formula: see text] the quotient group of [Formula: see text] by its center. The one-divisor graph of [Formula: see text], written as [Formula: see text], is defined to be a directed graph with [Formula: see text] as vertex set, and there is a directed edge from [Formula: see text] to [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text] and [Formula: see text] are, respectively, a left divisor and a right divisor of a rank one matrix in [Formula: see text]. The definition of [Formula: see text] is motivated by the definition of zero-divisor graph [Formula: see text] of [Formula: see text], which has vertex set of all nonzero zero-divisors in [Formula: see text] and there is a directed edge from a vertex [Formula: see text] to a vertex [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text]. The automorphism group of zero-divisor graph [Formula: see text] of [Formula: see text] was recently determined by Wang [A note on automorphisms of the zero-divisor graph of upper triangular matrices, Lin. Alg. Appl. 465 (2015) 214–220.]. In this paper, we characterize the automorphism group of one-divisor graph [Formula: see text] of [Formula: see text], proving that [Formula: see text], where [Formula: see text] is the automorphism group of field [Formula: see text], [Formula: see text] is a direct product of some symmetric groups. Besides, an application of automorphisms of [Formula: see text] is given in this paper.

Based on the idea of “q—count” of certain subwords of a word and generalizing the notion of Parikh matrix of a word, the notion of Parikh q—matrix of a word over an ordered alphabet was introduced. On the other hand, with a two-dimensional picture array of symbols arranged in rows and columns, two kinds of upper triangular matrices, known as row and column Parikh matrices have also been introduced and investigated. Here combining these two kinds of matrices of a picture array, we introduce row/column Parikh q—matrix of an array, leading to the concept of q—ambiguity of a picture array. Results relating to q—ambiguity of picture arrays are derived in the context of these Parikh q—matrices of arrays.

We study properties determined by idempotents in the following families of matrix semigroups over a semiring [Formula: see text]: the full matrix semigroup [Formula: see text], the semigroup [Formula: see text] consisting of upper triangular matrices, and the semigroup [Formula: see text] consisting of all unitriangular matrices. Il’in has shown that (for [Formula: see text]) the semigroup [Formula: see text] is regular if and only if [Formula: see text] is a regular ring. We show that [Formula: see text] is regular if and only if [Formula: see text] and the multiplicative semigroup of [Formula: see text] is regular. The notions of being abundant or Fountain (formerly, weakly abundant) are weaker than being regular but are also defined in terms of idempotents, namely, every class of certain equivalence relations must contain an idempotent. Each of [Formula: see text], [Formula: see text] and [Formula: see text] admits a natural anti-isomorphism allowing us to characterise abundance and Fountainicity in terms of the left action of idempotent matrices upon column spaces. In the case where the semiring is exact, we show that [Formula: see text] is abundant if and only if it is regular. Our main interest is in the case where [Formula: see text] is an idempotent semifield, our motivating example being that of the tropical semiring [Formula: see text]. We prove that certain subsemigroups of [Formula: see text], including several generalisations of well-studied monoids of binary relations (Hall relations, reflexive relations, unitriangular Boolean matrices), are Fountain. We also consider the subsemigroups [Formula: see text] and [Formula: see text] consisting of those matrices of [Formula: see text] and [Formula: see text] having all elements on and above the leading diagonal non-zero. We prove the idempotent generated subsemigroup of [Formula: see text] is [Formula: see text]. Further, [Formula: see text] and [Formula: see text] are families of Fountain semigroups with interesting and unusual properties. In particular, every [Formula: see text]-class and [Formula: see text]-class contains a unique idempotent, where [Formula: see text] and [Formula: see text] are the relations used to define Fountainicity, but yet the idempotents do not form a semilattice.

We study the effects of the branching osp(1|2n)⊃gl(n) on a particular class of simple infinite-dimensional osp(1|2n)-modules L(p) characterized by a positive integer p. In the first part (Sec. III), we use combinatorial methods, such as Young tableaux and Young subgroups, to construct a new basis for L(p) that respects this branching, and we express the basis elements explicitly in two distinct ways: first, as monomials of negative root vectors of gl(n) acting on certain gl(n)-highest weight vectors in L(p) and then as polynomials in the generators of osp(1|2n) acting on a osp(1|2n)-lowest weight vector in L(p). In the second part (Sec. IV), we use extremal projectors and the theory of Mickelsson–Zhelobenko algebras to give new explicit constructions of raising and lowering operators related to the branching osp(1|2n)⊃gl(n). We use the raising operators to give new expressions for the elements of the Gel’fand–Zetlin basis for L(p) as monomials of operators from U(osp(1|2n)) acting on a osp(1|2n)-lowest weight vector in L(p). We observe that the Gel’fand–Zetlin basis for L(p) is related to the basis constructed earlier in this paper by a triangular transition matrix. We end this paper (Sec. V) with a detailed example treating the case n = 3.

Let F be a field and let n, m be positive integers with m ≥ 2 and n ≥ 3. Denote by M n (F) (respectively, S n (F) and T n (F)) the linear space of all n × n full (respectively, symmetric and triangular) matrices over F. For any matrix A ∈ M n (F), the symbol A ad denotes the adjoint matrix of A. Let be two subspaces of M n (F) such that whenever , an operator is called an additive adjoint preserver if φ(A + B) = φ(A) + φ(B) for any and φ(A)ad = φ(A^ad) for every . We first characterize the additive adjoint preservers from S n (F) (respectively, T n (F)) to M m (F). Thereby, all additive adjoint preservers from S n (F) (respectively, M n (F) and T n (F)) to S m (F) (respectively, M m (F) and T m (F)) are described. Moreover, it is shown that f is an additive adjoint preserver from M n (F) to S m (F) (respectively, T m (F)) if and only if f = 0.

In this paper, we introduce the concept of a triangular coefficient matrix ring and investigate the structure of its ideals.We then characterize the radicals of the ring R h [x]/x n for every positive integer n, where R h [x] denotes the Hurwitz polynomial ring and x n represents the ideal of this ring generated by x n .Furthermore, we explore several properties that are transferred between the base ring R and the matrix ring Hn(R), which is a proper subring of the triangular coefficient matrix ring.