Irredundant Bases Research Articles

Irredundant bases for the symmetric group

AbstractAn irredundant base of a group acting faithfully on a finite set is a sequence of points in that produces a strictly descending chain of pointwise stabiliser subgroups in , terminating at the trivial subgroup. Suppose that is or acting primitively on , and that the point stabiliser is primitive in its natural action on points. We prove that the maximum size of an irredundant base of is , and in most cases . We also show that these bounds are best possible.

IBIS soluble linear groups

Let G be a finite permutation group on An ordered sequence of elements of Ω is an irredundant base for G if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of G have the same cardinality, G is said to be an IBIS group. In this paper we give a classification of quasi-primitive soluble irreducible IBIS linear groups, and we also describe nilpotent and metacyclic IBIS linear groups and IBIS linear groups of odd order.

Irredundant bases for finite groups of Lie type

We prove that the maximum length of an irredundant base for a primitive action of a finite simple group of Lie type is bounded above by a function which is a polynomial in the rank of the group. We give examples to show that this type of upper bound is best possible.

A classification of finite primitive IBIS groups with alternating socle

Abstract Let 𝐺 be a finite permutation group on Ω. An ordered sequence ( ω 1 , … , ω ℓ ) (\omega_{1},\ldots,\omega_{\ell}) of elements of Ω is an irredundant base for 𝐺 if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of 𝐺 have the same cardinality, 𝐺 is said to be an IBIS group. Lucchini, Morigi and Moscatiello have proved a theorem reducing the problem of classifying finite primitive IBIS groups 𝐺 to the case that the socle of 𝐺 is either abelian or non-abelian simple. In this paper, we classify the finite primitive IBIS groups having socle an alternating group. Moreover, we propose a conjecture aiming to give a classification of all almost simple primitive IBIS groups.

On relational complexity and base size of finite primitive groups

In this paper we show that if $G$ is a primitive subgroup of $S_{n}$ that is not large base, then any irredundant base for $G$ has size at most $5 \log n$. This is the first logarithmic bound on the size of an irredundant base for such groups, and is best possible up to a small constant. As a corollary, the relational complexity of $G$ is at most $5 \log n+1$, and the maximal size of a minimal base and the height are both at most $5 \log n.$ Furthermore, we deduce that a base for $G$ of size at most $5 \log n$ can be computed in polynomial time.

Statistics for Sn acting on k-sets

We study the natural action of Sn on the set of k-subsets of the set {1,…,n} when 1≤k≤n2. For this action we calculate the maximum size of a minimal base, the height and the maximum length of an irredundant base.Here a base is a set with trivial pointwise stabilizer, the height is the maximum size of a subset with the property that its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset, and an irredundant base can be thought of as a chain of (pointwise) set-stabilizers for which all containments are proper.

Primitive permutation IBIS groups

Let G be a finite permutation group on Ω. An ordered sequence of elements of Ω, (ω1,…,ωt), is an irredundant base for G if the pointwise stabilizer G(ω1,…,ωt) is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of G have the same size we say that G is an IBIS group. In this paper we show that if a primitive permutation group is IBIS, then it must be almost simple, of affine-type, or of diagonal type. Moreover we prove that a diagonal-type primitive permutation groups is IBIS if and only if it is isomorphic to PSL(2,2f)×PSL(2,2f) for some f≥2, in its diagonal action of degree 2f(22f−1).

ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP

AbstractLet G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).

Equational spectrum of Hilbert varieties

AbstractWe prove that an equational class of Hilbert algebras cannot be defined by a single equation. In particular Hilbert algebras and implication algebras are not one-based. Also, we use a seminal theorem of Alfred Tarski in equational logic to characterize the set of cardinalities of all finite irredundant bases of the varieties of Hilbert algebras, implication algebras and commutative BCK algebras: all these varieties can be defined by independent bases of n elements, for each n > 1.

On the single-orbit conjecture for uncoverings-by-bases

Let G be a permutation group acting on a finite set W. An uncovering-by-bases (or UBB) for G is a set U of bases for G such that any r-subset of W is disjoint from at least one base in U, where r = b d 1 2 c, for d the minimum degree of G. The single-orbit conjecture asserts that for any finite permutation group G, there exists a UBB for G contained in a single orbit of G on its irredundant bases. We prove a case of this conjecture, for when G is k-transitive and has a base of size k+1. Furthermore, in the more restricted case when G is primitive and has a base of size 2, we show how to construct a UBB of minimum possible size.

Uncoverings-by-bases for base-transitive permutation groups

An uncovering-by-bases for a group G acting on a finite set ? is a set $$\mathcal{U}$$ of bases for G such that any r-subset of ? is disjoint from at least one base in $$\mathcal{U}$$ , where r is a parameter dependent on G. They have applications in the decoding of permutation groups when used as error-correcting codes, and are closely related to covering designs. We give constructions for uncoverings-by-bases for many families of base-transitive group (i.e. groups which act transitively on their irredundant bases), including a general construction which works for any base-transitive group with base size 2, and some more specific constructions for other groups. In particular, those for the groups GL (3,q) and AGL (2,q) make use of the theory of finite fields. We also describe how the concept of uncovering-by-bases can be generalised to matroid theory, with only minor modifications, and give an example of this.

Pseudo-models and propositional Horn inference

A well-known result is that the inference problem for propositional Horn formulae can be solved in linear time. We show that this remains true even in the presence of arbitrary (static) propositional background knowledge. Our main tool is the notion of a cumulated clause, a slight generalization of the usual clauses in Propositional Logic. We show that each propositional theory has a canonical irredundant base of cumulated clauses, and present an algorithm to compute this base.

On Primitive Cellular Algebras

First we define and study the exponentiation of a cellular algebra by a permutation group which is analogous to the corresponding operation (the wreath product in primitive action) in permutation group theory. Necessary and sufficient conditions for the resulting cellular algebra to be primitive and Schurian are given. This enables us to construct infinite series of primitive non-Schurian algebras. Also we define and study for cellular algebras the notion of a base which is similar to that for permutation groups. We present an upper bound for the size of an irredundant base of a primitive cellular algebra in terms of the parameters of its standard representation. This produces new upper bounds for the order of the automorphism group of such an algebra and in particular for the order of a primitive permutation group. Finally we generalize to 2-closed primitive algebras some classical theorems for primitive groups and show that the hypothesis for a primitive algebra to be 2-closed is essential.

An interpolation theorem for irredundant bases of closure structures

The notion of closure structures of finite rank is introduced and several closure structures familiar from algebra and logic are shown to be of finite rank. The following theorem is established: Let k be any natural number and C any closure structure of rank k + 2. If B is a finite base (generating set) of C and D is an irredundant (independent) base of C such that | B| + k < | D|, then there is an irredundant base E of C such that | B| < | E| < | B| + k.

Possible cardinalities of irredundant bases for finite closure structures

We indicate certain connections between the rank and cardinality of a finite closure structure, and the relative sizes of its irredundant bases. A class of examples is described which shows that in general our theorem can not be strengthened.