Presentations for subgroups of monoids
The author generalizes the Reidemeister-Schreier result for subgroup presentations to presentations of subgroups of monoids. For a subset \(X\subseteq S\) of a monoid \(S\) and an element \(s\in S\), \(Xs\) is a coset if there exists an element \(t\in S\) such that \(Xst=X\). The index \([S:X]\) of \(X\) (in \(S\)) is the number of cosets of \(X\) in \(S\). Now \(S\) acts on the set \(I\cup\{0\}\) by \(C_is=C_{is}\), where \(C_i\) (\(i\in I\), \(0\not\in I\)) are the cosets of \(X\), and \(C_0\) satisfies \(C_is=C_0\) if \(C_is\) is not a coset, and \(C_0s=C_0\). Let \(G\) be a subgroup (with identity \(e\)) of a monoid \(S\). Then a set of elements \(r_i,r_i'\in S\) with \(i\in I\) is a system of coset representatives if \(Gr_i=C_i\), \(gr_ir_i'=g\) for all \(i\in I\) and \(g\in G\) (and \(r_1=r_1'=1\)). It is proved that if \(S\) is generated by \(A\), then \(Y=\{er_iar_{ia}'\mid i\in I,\;a\in A,\;ia\neq 0\}\) generates \(G\) as a monoid. Using a system of coset representatives, a monoid presentation is constructed for a subgroup \(G\) from a presentation of the monoid \(S\). As a corollary we have that a subgroup of finite index in a finitely presented monoid is finitely presented. The author also considers applications to the group of units of a monoid, and to regular monoids with finitely many left and right ideals.
scientific article; zbMATH DE number 1356306