In this paper, we extend some results of D. Dolzan on finite rings to profinite rings, a complete classification of profinite commutative rings with a monothetic group of units is given. We also prove the metrizability of commutative profinite rings with monothetic group of units and without nonzero Boolean ideals. Using a property of Mersenne numbers, we construct a family of power $2^{{\aleph }_0}$ commutative non-isomorphic profinite semiprimitive rings with monothetic group of units.

We generalize the results by G.V. Triantafillou and B. Fine on $G$-disconnected simplicial sets. An existence of an injective minimal model for a complete $𝕀$-algebra is presented, for any $EI$-category $𝕀$. We then make use of the $EI$-category $𝒪(G,X)$ associated with a $G$-simplicial set $X$ to apply these results to the category of $G$-simplicial sets.Finally, we describe the rational homotopy type of a nilpotent $G$-simplicial set by means of its injective minimal model.