Higman's Theorem Research Articles

Word problems, embeddings, and free products of right-ordered groups with amalgamated subgroup

We use permutation groups to give necessary and sufficient conditions for the free product of right-ordered groups with amalgamated subgroup to be right orderable. We obtain several consequences answering previously posed problems and also prove the right-orderable analogues of the Higman Embedding Theorem and the Boone–Higman Theorem.

Nilpotency of Zinbiel Algebras

Zinbiel algebras are defined by the identity (a ? b) ? c = a?(b?c+c?b). We prove an analog of the Nagata--Higman theorem for Zinbiel algebras. We establish that every finite-dimensional Zinbiel algebra over an algebraically closed field is solvable. Every solvable Zinbiel algebra with solvability length N is a nil-algebra with nil-index 2N if p = char K = 0 or p > 2N ? 1. Conversely, every Zinbiel nil-algebra with nil-index N is solvable with solvability length N if p = 0 or p > N ? 1. Every finite-dimensional Zinbiel algebra over complex numbers is nilpotent, nil, and solvable.

Well quasi-orders and context-free grammars

Let G be a context-free grammar and let L be the language of all the words derived from any variable of G. We prove the following generalization of Higman's theorem: any division order on L is a well quasi-order on L. We also give applications of this result to some quasi-orders associated with unitary grammars.

Rings of matrix invariants in positive characteristic

Denote by R n, m the ring of invariants of m-tuples of n× n matrices ( m, n⩾2) over an infinite base field K under the simultaneous conjugation action of the general linear group. When char( K)=0, Razmyslov (Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974) 723) and Procesi (Adv. Math. 19 (1976) 306) established a connection between the Nagata–Higman theorem and the degree bound for generators of R n, m . We extend this relationship to the case when the base field has positive characteristic. In particular, we show that if 0<char( K))⩽ n, then R n, m is not generated by its elements whose degree is smaller than m. A minimal system of generators of R 2, m is determined for the case char( K)=2: it consists of 2 m + m−1 elements, and the maximum of their degrees is m. We deduce a consequence indicating that the theory of vector invariants of the special orthogonal group in characteristic 2 is not analogous to the case char( K)≠2. We prove that the characterization of the R n, m that are complete intersections, known before when char( K)=0, is valid for any infinite K. We give a Cohen–Macaulay presentation of R 2,4, and analyze the difference between the cases char( K)=2 and char( K)≠2.

On Better-Quasi-Ordering Countable Series-Parallel Orders

We prove that any infinite sequence of countable series-parallel orders contains an increasing (with respect to embedding) infinite subsequence. This result generalizes Laver's and Corominas' theorems concerning betterquasi-order of the classes of countable chains and trees. Let C be a class of structures and < an order on C. This class is well-quasiordered with respect to < if for any infinite sequence C1,, C2... , Ck, ... in C, there exist i < j such that Ci < Cj. An equivalent definition is: any subset of C has finitely many minimal elements with respect to <. We are mainly concerned here with binary relations, the order < being the embedding order. Thus, 1? < R' when the relation R is embedded in the relation R' (in other words, R is an induced relation of R'). One of the very first results concerning well-quasi-order is Higman's theorem ([5]): the class of finite linear orderings (or chains) labeled by a finite set is well-quasi-ordered with respect to embedding. More precisely, the objects of the class are finite chains whose elements are labeled by a finite set, and embedding means order-preserving and label-preserving injection. For example aabcc embeds into abcabcabc but not into cbacbacba. This result was extended by Kruskal to the class of finite trees [8]. Then Nash-Williams introduced two fundamental tools of the theory: the 'minimal bad sequence' which shortened greatly the proofs concerning well-quasi-order (wqo), and the 'better-quasi-order' (bqo), a strengthening of wqo, which provides a tool to deal with countable structures [11]. Indeed, wqo is no longer an appropriate tool in the infinite case, since one can construct, from Rado's counterexample [13], an infinite set of pairwise incomparable countable subsets of a wqo. Laver proved in [9] that the class of countable chains is better-quasi-ordered (and thus well-quasi-ordered) with respect to einbedding. Later, Corominas [1] extended the result of Kruskal to countable trees: the class of countable trees labeled by a better-quasi-order is better-quasi-ordered with respect to embedding. The proof is in two parts, one devoted to the construction of countable trees, the other concerned with the preservation of better-quasi-order under certain operations. This latter aspect can be found in Milner [10] and Pouzet [12]. In this paper, Pouzet poses the problem of another class of orders, the N-free or series-parallel orders. This class contains the class of trees, and Pouzet was able to prove the wqo Received by the editors October 4, 1995 and, in revised form, January 6, 1998. 1991 Mathematics Subject Classification. Primary 05C20; Secondary 05C05,08A65,05C75.

Ordinal recursive bounds for Higman's theorem

The present paper concerns Higman's theorem for strings generated over a finite alphabet. We give a constructive proof of this theorem and we construct and characterise functions which bound the lengths of bad sequences. These bounding functions are described by ordinal-recursive definitions and their characterisation is achieved with reference to Hardy hierarchies of numbertheoretic functions.

Analogue of G. Higman's Theorem for Commutative Torsion-Free Rings

In this paper we consider finite rank torsion-free rings, which have almost regular automorphisms of prime order (a non-trivial automorphism is called almost regular if it has only trivial fixed points, i.e. zero and the elements of a ring linear dependent on its identity). The main result of this paper is the analogue of G. Higman's known Theorem [1] on almost regular automorphism for commutative finite rank torsion-free rings.

Higman's Embedding Theorem in a General Setting and Its Application to Existentially Closed Algebras

For a quasi variety of algebras K, the Higman Theorem is said to be true if every recursively presented K-algebra is embeddable into a finitely presented K-algebra; the Generalized Higman Theorem is said to be true if any K-algebra which is recursively presented over its finitely generated subalgebra is embeddable into a K-algebra which is finitely presented over this subalgebra. We suggest certain general conditions on K under which (1) the Higman Theorem implies the Generalized Higman Theorem; (2) a finitely generated K-algebra A is embeddable into every existentially closed K-algebra containing a finitely generated K-algebra B if and only if the word problem for A is Q-reducible to the word problem for B. The quasi varieties of groups, torsion-free groups, and semigroups satisfy these conditions.

Une généralisation des théorèmes de Higman et de Simon aux mots infinis

We prove that from an infinite sequence of finite or infinite words on a finite alphabet one can extract an increasing subsequence for the quasi-order ‘be a subword of’; that is, the extension of Higman's theorem to infinite words. Then we generalize Simon's theorem for infinite words.

Flag-Transitive Subgroups of Chevalley Groups

Publisher Summary Let G be a finite group of Chevalley type and B the Borel subgroup of G. If L ≤ G and G = BL, then L is said to be flagtransitive. D. Higman determined all flag-transitive subgroups of G in case G is of type A n . This chapter presents an extension of Higman's theorem to the case of a general group of Chevalley type. The result is used to show the nonexistence of 2-transitive permutation representations of certain groups of Chevalley type.