Nonstandard varieties of pseudotopological algebraic systems (Q1177452)

The author considers an algebraic system of arbitrary signature defined on a ground set \(A\), where \(A\) carries a pseudotopological convergence structure. This convergence structure is compatible with the algebra in the sense that all \(n\)-ary operators are continuous and all \(n\)-ary relations in \(A^ n\) are closed. Using nonstandard characterizations for such convergence structures, the author slightly develops the nonstandard theory of such pseudotopological algebraic systems and, in particular, extends a well-known theorem of Birkhoff to varieties of such structures. The author shows that if \(\mathcal H\) is a multiplicatively closed class of such structures, then (a) \(\mathcal H\) is a variety iff it is closed under closed subsystems and homomorphic images, (b) \(\mathcal H\) is a standard variety iff it is closed under subsystems and homomorphic images, (c) \(\mathcal H\) is a semistandard variety iff it is closed under subsystems and continuous homomorphic images, (d) \(\mathcal H\) is a topological variety iff it is closed under closed subsystems and continuous images. \{Reviewer's remark: This paper seems to give the impression that the author is the founder of the nonstandard theory of convergence spaces. The author has not included or mentioned either the first paper on this subject by \textit{C. Puritz} [``Quasimonad spaces: a nonstandard approach to convergence'', Proc. Lond. Math. Soc., III. Ser. 32, 230-250 (1976; Zbl 0319.54033)] nor the second paper on this subject by this reviewer [``A nonstandard approach to pseudotopological compactifications'', Z. Math. Logik Grundlagen Math. 26, 361-384 (1980; Zbl 0489.54020)]. The readers of this paper under review would benefit from first consulting these two previous foundational papers\}.