Crossed product Leavitt path algebras

If [Formula: see text] is a directed graph and [Formula: see text] is a field, the Leavitt path algebra [Formula: see text] of [Formula: see text] over [Formula: see text] is naturally graded by the group of integers [Formula: see text] We formulate properties of the graph [Formula: see text] which are equivalent with [Formula: see text] being a crossed product, a skew group ring, or a group ring with respect to this natural grading. We state this main result so that the algebra properties of [Formula: see text] are also characterized in terms of the pre-ordered group properties of the Grothendieck [Formula: see text]-group of [Formula: see text]. If [Formula: see text] has finitely many vertices, we characterize when [Formula: see text] is strongly graded in terms of the properties of [Formula: see text] Our proof also provides an alternative to the known proof of the equivalence [Formula: see text] is strongly graded if and only if [Formula: see text] has no sinks for a finite graph [Formula: see text] We also show that, if unital, the algebra [Formula: see text] is strongly graded and graded unit-regular if and only if [Formula: see text] is a crossed product. In the process of showing the main result, we obtain conditions on a group [Formula: see text] and a [Formula: see text]-graded division ring [Formula: see text] equivalent with the requirements that a [Formula: see text]-graded matrix ring [Formula: see text] over [Formula: see text] is strongly graded, a crossed product, a skew group ring, or a group ring. We characterize these properties also in terms of the action of the group [Formula: see text] on the Grothendieck [Formula: see text]-group [Formula: see text]

Real subset sums and posets with an involution

In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let [Formula: see text] be a finite poset, where [Formula: see text] is an order-reversing and involutive map such that [Formula: see text] for each [Formula: see text]. Let [Formula: see text] be the Boolean lattice with two elements and [Formula: see text] the family of all the order-preserving 2-valued maps [Formula: see text] such that [Formula: see text] if [Formula: see text] for all [Formula: see text]. In this paper, we build a family [Formula: see text] of particular subsets of [Formula: see text], that we call [Formula: see text]-bases on [Formula: see text], and we determine a bijection between the family [Formula: see text] and the family [Formula: see text]. In such a bijection, a [Formula: see text]-basis [Formula: see text] on [Formula: see text] corresponds to a map [Formula: see text] whose restriction of [Formula: see text] to [Formula: see text] is the smallest 2-valued partial map on [Formula: see text] which has [Formula: see text] as its unique extension in [Formula: see text]. Next we show how each [Formula: see text]-basis on [Formula: see text] becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.

Reduced finitary incidence algebras and their automorphisms

Let [Formula: see text] denote the incidence algebra of a locally finite poset [Formula: see text] over a field [Formula: see text] and [Formula: see text] some equivalence relation on the set of generators of [Formula: see text]. Then [Formula: see text] is the subset of [Formula: see text] of all the elements that are constant on the equivalence classes of [Formula: see text]. If [Formula: see text] satisfies certain conditions, then [Formula: see text] is a subalgebra of [Formula: see text] called a reduced incidence algebra. We extend this notion to finitary incidence algebras [Formula: see text] for any poset [Formula: see text]. We investigate reduced finitary incidence algebras [Formula: see text] and determine their automorphisms in some special cases.

Minimal degree of standard identities of matrix algebras with symplectic graded involution

Let [Formula: see text] be a field of characteristic zero and [Formula: see text] the algebra of [Formula: see text] matrices over [Formula: see text]. By the classical Amitsur–Levitzki theorem, it is well known that [Formula: see text] is the smallest degree of a standard polynomial identity of [Formula: see text]. A theorem due to Rowen shows that when the symplectic involution [Formula: see text] is considered, the standard polynomial of degree [Formula: see text] in symmetric variables is an identity of [Formula: see text]. This means that when only certain kinds of matrices are considered in the substitutions, the minimal degree of a standard identity may not remain being the same. In this paper, we present some results about the minimal degree of standard identities in skew or symmetric variables of odd degree of [Formula: see text] in the symplectic graded involution case. Along the way, we also present the minimal total degree of a double Capelli polynomial identity in symmetric variables of [Formula: see text] with transpose involution.

The lattice and semigroup structure of multipermutations

We study the algebraic properties of binary relations whose underlying digraph is smooth, that is, has no source or sink. Such objects have been studied as surjective hyper-operations (shops) on the corresponding vertex set, and as binary relations that are defined everywhere and whose inverse is also defined everywhere. In the latter formulation, they have been called multipermutations. We study the lattice structure of sets (monoids) of multipermutations over an [Formula: see text]-element domain. Through a Galois connection, these monoids form the algebraic counterparts to sets of relations closed under definability in positive first-order logic without equality. We show one side of this Galois connection, and give a simple dichotomy theorem for the evaluation problem of positive first-order logic without equality on the class of structures whose preserving multipermutations form a monoid closed under inverse. These problems turn out either to be in [Formula: see text]or to be [Formula: see text]-complete. We go on to study the monoid of all multipermutations on an [Formula: see text]-element domain, under usual composition of relations. We characterize its Green relations, regular elements and show that it does not admit a generating set that is polynomial on [Formula: see text].

Hierarchy for groups acting on hyperbolic ℤn-spaces

In [A.-P. Grecianu, A. Kvaschuk, A. G. Myasnikov and D. Serbin, Groups acting on hyperbolic [Formula: see text]-metric spaces, Int. J. Algebra Comput. 25(6) (2015) 977–1042], the authors initiated a systematic study of hyperbolic [Formula: see text]-metric spaces, where [Formula: see text] is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case [Formula: see text] taken with the right lexicographic order and studies the structure of finitely generated groups acting on hyperbolic [Formula: see text]-metric spaces. Under certain constraints, the structure of such groups is described in terms of a hierarchy (see [D. T. Wise, The Structure of Groups with a Quasiconvex Hierarchy[Formula: see text][Formula: see text]AMS-[Formula: see text], Annals of Mathematics Studies (Princeton University Press, 2021)]) similar to the one established for [Formula: see text]-free groups in [O. Kharlampovich, A. G. Myasnikov, V. N. Remeslennikov and D. Serbin, Groups with free regular length functions in [Formula: see text], Trans. Amer. Math. Soc. 364 (2012) 2847–2882].

Numerical upper bounds on growth of automaton groups

The growth of a finitely generated group is an important geometric invariant which has been studied for decades. It can be either polynomial, for a well-understood class of groups, or exponential, for most groups studied by geometers, or intermediate, that is between polynomial and exponential. Despite recent spectacular progresses, the class of groups with intermediate growth remains largely mysterious. Many examples of such groups are constructed using Mealy automata. The aim of this paper is to give an algorithmic procedure to study the growth of such automaton groups, and more precisely to provide numerical upper bounds on their exponents. Our functions retrieve known optimal bounds on the famous first Grigorchuk group. They also improve known upper bounds on other automaton groups and permitted us to discover several new examples of automaton groups of intermediate growth. All the algorithms described are implemented in GAP, a language dedicated to computational group theory.

Isomorphisms and derivations of partial flag incidence algebras

Let [Formula: see text] and [Formula: see text] be finite posets and [Formula: see text] a commutative unital ring. In the case where [Formula: see text] is indecomposable, we prove that the [Formula: see text]-linear isomorphisms between partial flag incidence algebras [Formula: see text] and [Formula: see text] are exactly those induced by poset isomorphisms between [Formula: see text] and [Formula: see text]. We also show that the [Formula: see text]-linear derivations of [Formula: see text] are trivial.

A Cayley graph for F2 × F2 which is not minimally almost convex

We give an example of a Cayley graph [Formula: see text] for the group [Formula: see text] which is not minimally almost convex (MAC). On the other hand, the standard Cayley graph for [Formula: see text] does satisfy the falsification by fellow traveler property (FFTP), which is strictly stronger. As a result, any Cayley graph property [Formula: see text] lying between FFTP and MAC (i.e., [Formula: see text]) is dependent on the generating set. This includes the well-known properties FFTP and almost convexity, which were already known to depend on the generating set as well as Poénaru’s condition [Formula: see text] and the basepoint loop shortening property (LSP) for which dependence on the generating set was previously unknown. We also show that the Cayley graph [Formula: see text] does not have the LSP, so this property also depends on the generating set.