On the fundamental double four-spiral semigroup (Q1914926)

The double four-spiral semigroup \(DSp_4\) is the semigroup freely generated by the idempotents \(a\), \(b\), \(c\), \(d\) and \(e\) subject to the defining relations that express the fact that \(a{\mathcal R}b{\mathcal L}c{\mathcal R}d{\mathcal L}e\leq a\). The author represents \(DSp_4\) as a semigroup of quadruples \((r,X;y,s)\) where \(r,s\in\{0,1\}\), \(y\) is a nonnegative integer and \(X\) belongs to a free monoid on two generators. The quadruples of the form \((0,X;y,0)\) form a subsemigroup \(\overline {A}\) and one can view the semigroup \(DSp_4\) as a Rees matrix semigroup over \(\overline {A}\). It is shown that if \(\theta\) is a congruence on \(DSp_4\) which is not contained in \(\mathcal L\), then \(DSp_4/\theta\) is completely simple.