Iterative Algebras without Projections

K. L. Safin ; E. V. Sukhanov

Algebra i logika, Tome 42 (2003) no. 1, pp. 107-122

Résumé

We deal with iterative algebras of functions of $k$-valued logic lacking projections, which we call algebras without projections. It is shown that a partially ordered set of algebras of functions of $m$-valued logic, for $m>k$, without projections contains an interval isomorphic to the lattice of all iterative algebras of functions of $k$-valued logic. It is found out that every algebra without projections is contained in some maximal algebra without projections, which is the stabilizer of a semigroup of non-surjective transformations of the basic set. It is proved that the stabilizer of a semigroup of all monotone non-surjective transformations of a linearly ordered 3-element set is not a maximal algebra without projections, but the stabilizer of a semigroup of all transformations preserving an arbitrary non one-element subset of the basic set is.