GLOBALS OF PSEUDOVARIETIES OF COMMUTATIVE SEMIGROUPS: THE FINITE BASIS PROBLEM, DECIDABILITY AND GAPS

Abstract Whereas pseudovarieties of commutative semigroups are known to be finitely based, the globals of monoidal pseudovarieties of commutative semigroups are shown to be finitely based (or of finite vertex rank) if and only if the index is 0, 1 or $\omega$. Nevertheless, on these pseudovarieties, the operation of taking the global preserves decidability. Furthermore, the gaps between many of these globals are shown to be big in the sense that they contain chains which are order isomorphic to the reals. AMS 2000 Mathematics subject classification: Primary 20M07; 20M05

[1]  Peter R. Jones,et al.  Locality of DS and associated varieties , 1995 .

[2]  Denis Thérien,et al.  Graph congruences and wreath products , 1985 .

[3]  Jorge Almeida,et al.  On finitely based pseudovarieties of the forms V ∗ D and V ∗ Dn , 2000 .

[4]  Pascal Weil,et al.  Relatively Free Profinite Monoids: An Introduction and Examples , 1995 .

[5]  J. Almeida.  Some Order Properties of the Lattice of Varieties of Commutative Semigroups , 1986, Canadian Journal of Mathematics.

[6]  Jorge Almeida.  Hyperdecidable Pseudovarieties and the Calculation of Semidirect Products , 1999, Int. J. Algebra Comput..

[7]  Howard Straubing,et al.  FINITE SEMIGROUP VARIETIES OF THE FORM V,D , 1985 .

[8]  Jorge Almeida,et al.  Finite Semigroups and Universal Algebra , 1995 .

[9]  Jorge Almeida,et al.  The Gap Between Partial and Full , 1998, Int. J. Algebra Comput..

[10]  Jorge Almeida,et al.  On the Decidability of Iterated Semidirect Products with Applications to Complexity , 2000 .

[11]  Janusz A. Brzozowski,et al.  Characterizations of locally testable events , 1971, Discret. Math..

[12]  Jorge Almeida.  Semidirect products of pseudovarieties from the universal algebraist's point of view , 1989 .

[13]  Bret Tilson,et al.  Categories as algebra: An essential ingredient in the theory of monoids , 1987 .

[14]  Robert Knast.  Some Theorems on Graph Congruences , 1983, RAIRO Theor. Informatics Appl..

[15]  Jorge Almeida,et al.  Syntactic and Global Semigroup Theory: A Synthesis Approach , 2000 .

[16]  Pascal Weil,et al.  Profinite categories and semidirect products , 1998 .

[17]  E. Nelson.  The Lattice of Equational Classes of Commutative Semigroups , 1971, Canadian Journal of Mathematics.