THE p-COCKCROFT PROPERTY OF THE SEMI-DIRECT PRODUCTS OF MONOIDS

Abstract

The semi-direct product of arbitrary two monoids and a presentation for this product have received considerable attention, see for instance [12, 14, 15]. In [15], Wang defined a trivializer set of the Squier complex associated with this presentation. In this paper, as a main result, we discuss necessary and sufficient conditions for the standard presentation of the semi-direct product of any two monoids to be p-Cockcroft for any prime p or 0. Finally we present some applications of this main theorem.

A New Example of Deficiency One Groups

“…By [2], each of X A and X D contains a single generating picture P A and P D , respectively as drawn in Figure 2 in [4]. By [2], each of X A and X D contains a single generating picture P A and P D , respectively as drawn in Figure 2 in [4].…”

“…Clearly to obtain a spherical picture, say P sc , from this last non-spherical picture, we must combine a and a −1 by an arc (see Figure 3-(a) in [4]). If we process the boundary of B s,c by a single a-arc, then for each fixed y ∈ {s, c}, we get one positive and one negative T ya -discs.…”

“…Now let us consider the picture P sc as drawn in Figure 3-(a) in [4]. In other words, in the proof, we will basically count the number of discs in each of spherical pictures P A , P D , P sc and P ya , where y ∈ {s, c}.…”

“…For a fixed y ∈ {s, c}, let us consider a picture P ya (see Figure 3-(b) in [4]). It contains one positive and one negative a t -discs and two subpictures A a t ,y and B y,a t , where y ∈ {s, c}.…”

“…By considering Figures 1, 2 and 3 in [4], one can easily draw the generating pictures for π 2 (P 1 ) while k = 1. Assume k = 1 in P 1 .…”