Post hulls of spaces of signatures. (Q5932590)

For an arbitrary Boolean space \(K\) and an even number \(n\) let \(G_K=C(K,\mu_n)\) be the set of all continuous functions from \(K\) to the group of the complex \(n\)-th roots of unity with the natural group structure. The group \(G_K\) can be considered as a Post algebra of order \(n\). Every element \(x\in K\) determines its evaluation map \(e_x\in G^*_K.\) The author proves that if \(E_K=\{e_x^l\); \(x\in K\), \(l\) an odd integer\} and \(-1\in G_K\) denotes the constant function, then the triple \((E_K ,G_K,-1)\) is a space of signatures in the meaning of \textit{C. Mulcahy} [Commun. Algebra 16, No. 3, 577--612 (1988; Zbl 0644.10016) and J. Algebra 119, No. 1, 105--122 (1988; Zbl 0659.10020)]. Thus for any space of signatures \((X,G,-1)\) one can construct a new space of signatures \((E_X,G_X,-1)\) which is called the Post hull of \((X,G,-1).\) Moreover, there exists a natural homomorphism \(\varepsilon:(X,G,-1)\to(E_X,G_X,-1).\) The author shows that the pair \(((E_X,G_X,-1),\varepsilon)\) satisfies a certain universal property. The construction of the Post hull of the space of signatures presented in the paper is a generalization of the construction of the Boolean hull of the space of orderings invented by \textit{M. Dickmann} and \textit{F. Miraglia} [Mem. Am. Math. Soc. 689 (2000; Zbl 1052.11027)].