Varieties of topological algebras

By a variety of topological algebras we mean a class V of topological algebras of a fixed type closed under the formation of subalgebras, products and quotients (i.e. images under continuous homomorphisms yielding the quotient topology). In symbols, V = SV = PV = QV. if V is also closed under the formation of arbitrary continuous homomorphic images, then V is a wide variety . variety. As an example we have the full variety V = Mod r (Σ), the class of all topological algebras of a fixed type τ obeying a fixed set Σ of algebraic identities. But not every wide variety is full, e.g. the class of all indiscrete topological algebras of a fixed type; in fact, as Morris observed (1970b), there exists a proper class of varieties of topological groups.

[1]  Reflexive Homomorphic Relations , 1960, Canadian Mathematical Bulletin.

[2]  A. Robinson.  Non-standard analysis , 1966 .

[3]  Walter Taylor,et al.  Residually small varieties , 1972 .

[4]  Bjarni Jónsson.  On the representation of lattices , 1953 .

[5]  S. Swierczkowski,et al.  Topologies in Free Algebras , 1964 .

[6]  W. Taylor.  Some constructions of compact algebras , 1971 .

[7]  W. Taylor.  Varieties Obeying Homotopy Laws , 1977, Canadian Journal of Mathematics.

[8]  V. Kannan.  Reflexive cum coreflexive subcategories in topology , 1971 .

[9]  J. Gerhard.  Some subdirectly irreducible idempotent semigroups , 1972 .

[10]  Walter D. Neumann,et al.  On Malcev conditions , 1974, Journal of the Australian Mathematical Society.

[11]  R. P. Dilworth,et al.  Algebraic theory of lattices , 1973 .

[12]  W. Taylor.  Uniformity of congruences , 1974 .

[13]  J. M. Boardman,et al.  Homotopy Invariant Algebraic Structures on Topological Spaces , 1973 .

[14]  Walter Taylor,et al.  Characterizing Mal’cev conditions , 1973 .

[15]  VARIETIES OF BANACH ALGEBRAS , 1976 .

[16]  S. Morris.  Varieties of topological groups , 1969, Bulletin of the Australian Mathematical Society.

[17]  W. Taylor.  Pure Compactifications in Quasi-Primal Varieties , 1976, Canadian Journal of Mathematics.

[18]  S. Morris,et al.  Generating varieties of topological groups , 1973, Proceedings of the Edinburgh Mathematical Society.

[19]  STEPHrN L. BLOOM,et al.  Varieties of Ordered Algebras , 1976, J. Comput. Syst. Sci..

[20]  Wolfgang M. Schmidt.  Approximation to algebraic numbers , 1972 .

[21]  Compactness as a type of injectivity , 1974 .

[22]  The class of topological spaces is equationally definable , 1973 .

[23]  George Grätzer,et al.  Two Mal'cev-type theorems in universal algebra , 1970 .

[24]  J. Diestel,et al.  Varieties of linear topological spaces , 1972 .