We prove that monoids Mon〈a, b, c, d : ab = 0, ac = 1, db = 1, dc = 1 dab = 1, dab = 1, . . . , dab = 1〉 are congruence-free for all n ≥ 1. This provides a new countable family of finitely presented congruence-free monoids, bringing one step closer to understanding the Boone–Higman Conjecture. We also provide examples which show that finitely presented congruence-free monoids may have quadratic...

2. G. Higman, On finite groups of exponent five, Proc. Cambridge Philos. Soc. 52 (1956), 381-390. 3. A. I. Kostrikin, On Burnside's problem, Dokl. Akad. Nauk SSSR 119 (1958), 1081-1084. (Russian) 4. M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. École Norm. Sup. (3) 71 (1954), 101-190. 5. H. Zassenhaus, Ein Verfahren, jeder endlichen p-Gruppe eine Lie-Ring mit der Charakteris...

In this paper we establish some subnormal embeddings of groups into groups with additional properties; in particular embeddings of countable groups into 2-generated groups with some extra properties. The results obtained are generalizations of theorems of P. Hall, R. Dark, B. Neumann, Hanna Neumann, G. Higman on embeddings of that type. Considering subnormal embeddings of finite groups into fin...

This paper describes how to verify cryptographic protocols by a general-purpose program transformation technique with unfolding. The questions of representation and analysis of the protocols as prefix rewriting grammars are discussed. In these aspects Higman and Turchin embeddings on computational paths are considered, and a refinement of Turchin’s relation is presented that allows to algorithm...

The structure theory for Pi-algebras is well developed. Some results of this theory are classic now. One of them is Kaplansky's theorem which asserts that a primitive Pi-algebra is finite dimensional over its centre. Another example is the theorem of Nagata-Higman which asserts that any algebra over a field of zero characteristic satisfying identity x" = 0 is nilpotent. In 1957 A.I. Shirshov pr...

In this note, we sharpen work of Ulas to provide what is, in some sense, the minimal counterexample to a “conjecture” of Erdős and Graham about square values of products of disjoint blocks of consecutive integers.

Many famous graphs are edge-primitive, for example, the Heawood graph, the Tutte–Coxeter graph and the Higman–Sims graph. In this paper we systematically analyse edge-primitive and edge-quasiprimitive graphs via the O’Nan–Scott Theorem to determine the possible edge and vertex actions of such graphs. Many interesting examples are given and we also determine all G-edge-primitive graphs for G an ...

We construct non-periodic 2-colorings that avoid long monochromatic progressions having odd common differences. Also we prove that the set of all arithmetic progressions with common differences in (N!−1) ∪ N! ∪ (N!+1)− {0} does not have the 2-Ramsey property.

In this paper we relate the Fefferman–Graham ambient metric construction for conformal manifolds to the approach to conformal geometry via the canonical Cartan connection. We show that from any ambient metric that satisfies a weakening of the usual normalisation condition, one can construct the conformal standard tractor bundle and the normal standard tractor connection, which are equivalent to...

It is conjectured by Erdős, Graham and Spencer that if 1 ≤ a1 ≤ a2 ≤ · · · ≤ as with ∑s i=1 1/ai < n − 1/30, then this sum can be decomposed into n parts so that all partial sums are ≤ 1. In this note we propose a counterexample which gives a negative answer to this conjecture.