On locally finite Menger algebras
by
Jānis Cīrulis
University of Latvia

Let a be any ordinal. A Menger algebra of dimension a, or an a-clone, is an algebra (W, ○, e_i) _{i < a}, where ○ is a (1+a)-ry operation on W, every e_i is an element of W, and the following axioms hold (boldface letters denote a-tuples from W^a; in particular, e stands for (e₀, e₁, ...)):
       w ○e = w,     e_i ○v = v_i,     w ○(u *v) = (w ○u) ○v,
the tuple u *v being defined pointwise by (u *v)_i = u_i ○v.

In the talk, we shall deal with w-clones. An w-clone W is said to be locally finite-dimensional (for short: locally finite) if to every w ∈ W there is a natural number n such that w ○u = w whenever u_i = e_i for all i < n. We discuss interrelations between w-clones, on the one hand, and relatively free algebras, clones and varieties, on the other. In particular, the categories of clones and of locally finite w-clones are equivalent.

Date received: May 9, 2008