An ordered pair X (R) = (X, R ) consisting of a nonvoid set X and a nonvoid family R of binary relations on X is called a relator space. Relator spaces are straightforward generalizations not only of uniform spaces, but also of ordered sets. Therefore, in a relator space we can naturally define not only some topological notions, but also some order theoretic ones. It turns out that these two, a...

It is known that the word problem for one-relator groups and for one-relator monoids of the form Mon〈A ‖ w = 1〉 is decidable. However, the question of decidability of the word problem for general one-relation monoids of the form M = Mon〈A ‖ u = v〉 where u and v are arbitrary (positive) words in A remains open. The present paper is concerned with one-relator inverse monoids with a presentation o...

Abstract Languages can be divided into three types with respect to the encoding of comitatives and instrumentals: identity, differentiation mixed (Stolz, Stroh Urdze 2013). Diachronic data from Thai dating 13th 21st centuries ce suggests that these language correspond stages development relation between two categories in Thai, which progress as follows. Between mid-18th centuries, employed patt...

The statement “no nonabelian simple group can be obtained from a nonsimple group by adding one generator and one relator” 1) is equivalent to the Kervaire–Laudenbach conjecture; 2) becomes true under the additional assumption that the initial nonsimple group is either finite or torsion-free.

We prove that the natural map G → Ĝ, where G is a torsionfree group and Ĝ is obtained by adding a new generator t and a new relator w , is surjective only if w is conjugate to gt where g ∈ G . This solves a special case of the surjectivity problem for group extensions, raised by Cohen [2]. AMS Classification 20E22, 20F05; 57M20, 57Q10

A relative one-relator presentation has the form P = 〈x,H;R〉 where x is a set, H is a group, and R is a word on x±1 ∪H. We show that if the word on x±1 obtained from R by deleting all the terms from H has what we call the unique max-min property, then the group defined by P is residually finite if and only if H is residually finite (Theorem 1). We apply this to obtain new results concerning the...

We prove that Whitehead’s algorithm for solving the automorphism problem in a fixed free group Fk has strongly linear time generic-case complexity. This is done by showing that the “hard” part of the algorithm terminates in linear time on an exponentially generic set of input pairs. We then apply these results to one-relator groups. We obtain a Mostow-type isomorphism rigidity result for random...

When two nominal constituents are in a relation of dependency in the Bantu languages, the syntactically dependent constituent is usually introduced by a relator. In Bantu studies this relator is called connective (also associative, genitive or connexive) element (or pronoun, clitic, prefix, marker or particle). The order is HEAD (R1) RELATOR (REL) DEPENDENT (R2). R1 and R2 are short for first a...

In this paper, we generalise Magnus’ Freiheitssatz and solution to the word problem for one-relator groups by considering one relator quotients of certain classes of right-angled Artin groups and graph products of locally indicable polycyclic groups.