Narrow Results

We consider birth-and-death stochastic particle systems in continuum which are under a self-regulation mechanism controlling configurations of particles via a pairwise interaction between them. The latter is reflected in a potential perturbation of the free generator. We show that the ground state renormalization scheme in the considered model leads to an invariant measure, a renormalized generator and resulting equilibrium birth-and-death stochastic dynamics for the system. The proof is based on the Gibbs-type representation for related path space measure. This measure has OS-positivity property and is constructed via the cluster expansion method.

We consider the relationship between the combinatorial properties of semigroupoids in general and semigroups in particular. We show that a semigroupoid is finitely generated [finitely presentable] exactly if the corresponding categorical-at-zero semigroup is finitely generated [respectively, finitely presentable]. This allows us to extend some of the main results of [17], to show that finite generation and presentability are preserved under finite extension of semigroupoids and the taking of cofinite subsemigroupoids. We apply this result to extend the results of [6], giving characterizations of finite generation and finite presentability in Rees matrix semigroups over semigroupoids.

Given tuples a₁, …, a_k and b in Aⁿ for some algebraic structure A, the subpower membership problem asks whether b is in the subalgebra of Aⁿ that is generated by a₁, …, a_k. For A a finite group, there is a folklore algorithm which decides this problem in time polynomial in n and k. We show that the subpower membership problem for any finite Mal'cev algebra is in NP and give a polynomial time algorithm for any finite Mal'cev algebra with finite signature and prime power size that has a nilpotent reduct. In particular, this yields a polynomial algorithm for finite rings, vector spaces, algebras over fields, Lie rings and for nilpotent loops of prime power order.

Let 𝒫_X and 𝒮_X be the partition monoid and symmetric group on an infinite set X. We show that 𝒫_X may be generated by 𝒮_X together with two (but no fewer) additional partitions, and we classify the pairs α, β ∈ 𝒫_X for which 𝒫_X is generated by 𝒮_X ∪ {α, β}. We also show that 𝒫_X may be generated by the set ℰ_X of all idempotent partitions together with two (but no fewer) additional partitions. In fact, 𝒫_X is generated by ℰ_X ∪ {α, β} if and only if it is generated by ℰ_X ∪ 𝒮_X ∪ {α, β}. We also classify the pairs α, β ∈ 𝒫_X for which 𝒫_X is generated by ℰ_X ∪ {α, β}. Among other results, we show that any countable subset of 𝒫_X is contained in a 4-generated subsemigroup of 𝒫_X, and that the length function on 𝒫_X is bounded with respect to any generating set.

The variant of a semigroup $S$ with respect to an element $a\in S$, denoted $S^a$, is the semigroup with underlying set $S$ and operation ⋆ defined by $x\star y=xay$ for $x,y\in S$. In this paper, we study variants ${{𝒯}}_X^a$ of the full transformation semigroup ${{𝒯}}_X$ on a finite set $X$. We explore the structure of ${{𝒯}}_X^a$ as well as its subsemigroups $\mathrm{\text{Reg}}({{𝒯}}_X^a)$ (consisting of all regular elements) and ${{ℰ}}_X^a$ (consisting of all products of idempotents), and the ideals of $\mathrm{\text{Reg}}({{𝒯}}_X^a)$. Among other results, we calculate the rank and idempotent rank (if applicable) of each semigroup, and (where possible) the number of (idempotent) generating sets of the minimal possible size.

We consider the algebra of invariants of $d$-tuples of $n\times n$ matrices under the action of the orthogonal group by simultaneous conjugation over an infinite field of characteristic $p$ different from two. It is well known that this algebra is generated by the coefficients of the characteristic polynomial of all products of generic and transpose generic $n\times n$ matrices. We establish that in case $0<p\le n$ the maximal degree of indecomposable invariants tends to infinity as $d$ tends to infinity. In other words, there does not exist a constant $C(n)$ such that it only depends on $n$ and the considered algebra of invariants is generated by elements of degree less than $C(n)$ for any $d$. This result is well-known in case of the action of the general linear group. On the other hand, for the rest of $p$ the given phenomenon does not hold. We investigate the same problem for the cases of symmetric and skew-symmetric matrices.

The algebra of ${\mathrm{\text{GL}}}_n$-invariants of m-tuples of $n\times n$ matrices with respect to the action by simultaneous conjugation is a classical topic in case of infinite base field. On the other hand, in case of a finite field generators of polynomial invariants are not known even for a pair of $2\times 2$ matrices. Working over an arbitrary field we classified all ${\mathrm{\text{GL}}}_2$-orbits on m-tuples of $2\times 2$ nilpotent matrices for all $m>0$. As a consequence, we obtained a minimal separating set for the algebra of ${\mathrm{\text{GL}}}_2$-invariant polynomial functions of m-tuples of $2\times 2$ nilpotent matrices. We also described the least possible number of elements of a separating set for an algebra of invariant polynomial functions over a finite field.

The fundamental group of a 2-bridge knot has a particularly nice presentation, having only two generators and a single relation. For certain families of 2-bridge knots, such as the torus knots, or the twist knots, the relation takes on an especially simple form. Exploiting this form, we derive a formula for the A-polynomial of twist knots. Our methods extend to at least one other infinite family of (non-torus) 2-bridge knots. Using these formulae we determine the associated Newton polygons. We further prove that the A-polynomials of twist knots are irreducible.

A subnormal subgroup X of G has the following property: there exists a Dirichlet polynomial Q(s) with integer coefficients such that, for each t ∈ ℕ, Q(t) is the conditional probability that t random elements generate G given that they generate G together with the elements of X In this paper we analyze how far can a subgroup X be with this property from being a subnormal subgroup.

Let M be a Clifford monoid and let θ be an endomorphism of M. We prove that if the Bruck–Reilly extension BR(M, θ) is finitely presented then M is finitely generated. This allows us to derive necessary and sufficient conditions for Bruck–Reilly extensions of Clifford monoids to be finitely presented.

In this note, we consider the semigroup ${𝒪}(X)$ of all order endomorphisms of an infinite chain $X$ and the subset $J$ of ${𝒪}(X)$ of all transformations $\alpha$ such that $|\mathrm{\text{Im}}(\alpha )|=\left|X\right|$. For an infinite countable chain $X$, we give a necessary and sufficient condition on $X$ for ${𝒪}(X)=〈J〉$ to hold. We also present a sufficient condition on $X$ for ${𝒪}(X)=〈J〉$ to hold, for an arbitrary infinite chain $X$.

For $n\in \mathbb{N}$, let $X_n=\{a_1,a_2,\dots ,a_n\}$ be an $n$-element set and let $F=(X_n;{<}_f)$ be a fence, also called a zigzag poset. As usual, we denote by $I_n$ the symmetric inverse semigroup on $X_n$. We say that a transformation $\alpha \in I_n$ is fence-preserving if $x{<}_{f}y$ implies that $x\alpha {<}_{f}y\alpha$, for all $x,y$ in the domain of $\alpha$. In this paper, we study the semigroup $PF{I}_n$ of all partial fence-preserving injections of $X_n$ and its subsemigroup $I{F}_n=\{\alpha \in PF{I}_n:{\alpha }^{-1}\in PF{I}_n\}$. Clearly, $I{F}_n$ is an inverse semigroup and contains all regular elements of $PF{I}_n.$ We characterize the Green’s relations for the semigroup $I{F}_n$. Further, we prove that the semigroup $I{F}_n$ is generated by its elements with rank $\ge n-2$. Moreover, for $n\in 2\mathbb{N}$, we find the least generating set and calculate the rank of $I{F}_n$.

In this paper, we show that for any finite field $K$, any pair of map-generators (that is when one of the generators is an involution) of $\mathrm{\text{SL}}(3,K)$ and $\mathrm{\text{PSL}}(3,K)$ has a group automorphism that inverts both generators. In the theory of maps, this corresponds to say that any regular oriented map with automorphism group $\mathrm{\text{SL}}(3,K)$ or $\mathrm{\text{PSL}}(3,K)$ is reflexible, or equivalently, there are no chiral regular maps with automorphism group $\mathrm{\text{SL}}(3,K)$ or $\mathrm{\text{PSL}}(3,K)$. As remarked by Leemans and Liebeck, also $\mathrm{\text{SU}}(3,K)$ and $\mathrm{\text{PSU}}(3,K)$ are not automorphism groups of chiral regular maps. These two results complete the work of the above authors on simples groups supporting chiral regular maps.

The work of Helmer [Divisibility properties of integral functions, Duke Math. J. 6(2) (1940) 345–356] applied algebraic methods to the field of complex analysis when he proved the ring of entire functions on the complex plane is a Bezout domain (i.e. all finitely generated ideals are principal). This inspired the work of Henriksen [On the ideal structure of the ring of entire functions, Pacific J. Math. 2(2) (1952) 179–184. On the prime ideals of the ring of entire functions, Pacific J. Math. 3(4) (1953) 711–720] who proved a correspondence between the maximal ideals within the ring of entire functions and ultrafilters on sets of zeroes as well as a correspondence between the prime ideals and growth rates on the multiplicities of zeroes. We prove analogous results on rings of analytic functions in the non-Archimedean context: all finitely generated ideals in the ring of analytic functions on an annulus of a characteristic zero non-Archimedean field are two-generated but not guaranteed to be principal. We also prove the maximal and prime ideal structure in the non-Archimedean context is similar to that of the ordinary complex numbers; however, the methodology has to be significantly altered to account for the failure of Weierstrass factorization on balls of finite radius in fields which are not spherically complete, which was proven by Lazard [Les zeros d’une function analytique d’une variable sur un corps value complet, Publ. Math. l’IHES 14(1) (1942) 47–75].

This paper studies the finite groups G that have a pair (A,B) of non-trivial proper subgroups satisfying the condition that for every H ⩽ G either A ⩽ H or H ⩽ B.

Let M be a monoid and let θ be an endomorphism of M. We prove that if the Bruck–Reilly extension BR(M, θ) is finitely presented, then the Bruck–Reilly extension BR(M, θ^m) is also finitely presented for all m ⩾ 1.

Let X be one of the finite-dimensional graded simple Lie superalgebras of Cartan type W, S, H, K, HO, KO, SHO or SKO over an algebraically closed field of characteristic p > 3. In this paper we prove that X can be generated by one element except the ones of type W, HO, KO or SKO in certain exceptional cases in which X can be generated by two elements. As a subsidiary result, we prove that certain classical Lie superalgebras or their relatives can be generated by one or two elements.

In this paper, we describe a method to determine the structure of the Tarski algebra with a finite set of free generators which is different to that given by Iturrioz and Monteiro in [Les algèbres de Tarski avec un nombre fini de générateurs libres, in Informe Técnico, Vol. 37 (Instituto de Matemática de la Universidad Nacional del Sur, Bahía Blanca, 1994)].

In this paper, we consider endomorphisms of a finite directed path from monoid generators perspective. Our main aim is to determine the rank of the monoid $\mathrm{\text{wEnd}}{\overrightarrow{P}}_n$ of all weak endomorphisms of a directed path with $n$ vertices, which is a submonoid of the widely studied monoid ${{𝒪}}_n$ of all order-preserving transformations of an $n$-chain. Also, we describe the regular elements of $\mathrm{\text{wEnd}}{\overrightarrow{P}}_n$ and calculate its size and number of idempotents.

Let F_n be a free group of rank n with basis X = {x₁, x₂, ⋯, x_n}. If Y = {y₁, y₂, ⋯, y_n} is any other set of generators for F_n then they form a basis for F_n since finitely generated free groups are Hopfian. In this paper which shall show that this fact characterizes finitely generated free groups.