Products of inverse semigroups
Let \(S\) and \(T\) be inverse semigroups and let \(\theta:S\to\text{End }T\) be an antimorphism (which will remain fixed for the discussion, and whose action \(t(s\theta)\) is denoted simply as \(t^s\) for \(s\in S\) and \(t\in T\)). The standard semidirect product of \(S\) and \(T\), denoted here by \(T_{\theta\times_A}S\) has multiplication \((t,s)(t_1,s_1)=(tt^s_1,ss_1)\). To generalize this structure the paper considers systems \((T\times S,\theta,u,v)\) and multiplications \((t_1,s_1)(t_2,s_2)=(t^{u(s_1,s_2)}_1t^{v(s_1,s_2)}_2,s_1s_2),\) where \(u(x_1,x_2)\) and \(v(x_1,x_2)\) are words representing elements of the free inverse semigroup (with identity adjoined) on two generators, \(x_1\) and \(x_2\). The paper exhibits 8 sets of words \((u,v)\) which will produce associative products. These sets come in dual pairs, which means that there are 4 essentially different sets of words. These are \(\bullet\) \((1,1)\), which gives the direct product, \(\bullet\) \((1,x_1)\), which gives the standard semidirect product, \(\bullet\) \((x_1x_2x^{-1}_2x^{-1}_1,x_1)\), and \(\bullet\) \((x_1x_2x_2^{-1}x^{-1}_1,x_1x_2x^{-1}_2)\). By using special semigroups, in particular, free inverse semigroups, as factors in the products, the paper establishes equations which \(u\) and \(v\) must satisfy in order for any resulting product to be associative. Although the paper does not prove that these 8 pairs are the only solutions which will work, it reports that an exhaustive computer search has determined that they are the only words of length less than 12 which will work. This reviewer thinks that it is worth noting an alternate notation for the last two pairs above. Recalling that in an inverse semigroup an element \(s\) has an associated left identity element, \(e_s=ss^{-1}\) and simplifying the notation in the equations above by replacing ``\(x_1\)'' by ``\(x\)'' and replacing ``\(x_2\)'' by \(y\), one sees that they can be written as \((e_{xy},x)\) and \((e_{xy},xe_y)\), resp. (Noting that in an inverse semigroup \(xe_y=e_{xy}x\) makes this last pair even more symmetrical.) Since every idempotent \(e\) of \(S\) produces a retraction \(T\to T^e\) of \(T\) onto a subsemigroup of \(T\), it appears that these equations are very closely related to the idempotent structure of \(S\).
scientific article; zbMATH DE number 869952