A Description of Normal Semigroups of Endomorphisms of Proper Independence Algebras

Abstract

Let A be a strong independence algebra of finite rank with at most one constant, and let G be the group of automorphisms of A. Let α be a singular endomorphism of A and α G = {α} ∪ G . We describe the elements of α G and give additional characterisations when A is a proper independence algebra and G is a periodic group.Mathematics subject classification: 20M20, 20M10, 08A35.

v*-ALGEBRAS, INDEPENDENCE ALGEBRAS AND LOGIC

“…(1) aS = bS if and only if a and b have the same kernel; (2) Sa = Sb if and only if a and b have the same range; (3) SaS = SbS if and only if a and b have the same rank; (4) SaS = SbS if and only if there exists a c in S such that aS = cS and Sc = Sb; (5) the semigroup S \U is generated by the set of its idempotent elements when the dimension of the underlying algebra (V or X) is finite (see [24,48] and also [4,7,17,23,61]); (6) if a ∈ S \ U , then {a} ∪ U \ U is generated by the set of its idempotent elements when the dimension of the underlying algebra is finite (see [3,6,9,10]); (7) there exists an injective mapping a and a surjective mapping b in S such that S = U ∪ {a, b} when the dimension of the underlying algebra is infinite (see [2,8,46,47]); (8) the subsemigroup of S consisting of all elements of finite rank is a completely semisimple semigroup, that is to say, it is regular and its principal factors are all completely 0-simple or completely simple (see [41]). …”