**
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

Vol. 61, 1 (1992)

pp. 85-90

ON SURJECTIVE KERNELS OF PARTIAL ALGEBRAS

P. ZLATOS

**Abstract**.
A partial algebra $\A = (A,F)$ is called surjective if each of its elements lies in the range of some of its operations. By a transfinite iteration construction over the class of all ordinals it is proved that in each partial algebra $\A$ there exists the largest surjective subalgebra $\Skr \A$, called the surjective kernel of $\A$. However, what might be found a bit surprising, for each ordinal $\al$ there is an algebra $\A$ with only finitary operations (even with a single unary operation), such that the described construction stops exactly in $\al$ steps. The result is compared with the classical ones on perfect kernels of first countable topological spaces.

**AMS subject classification**.
08A05, 08A55, 08A62, 08A65; Secondary 08A30, 08A60, 05C05, 54A20

**Keywords**.
Partial algebra, subalgebra, infinitary partial operation, surjective kernel, tree, limit, accumulation point

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