## Kuratowski monoids of $$n$$-topological spaces.(English)Zbl 1398.54003

The 14-set closure-complement theorem of C. Kuratowski [Fundam. Math. 3, 182–199 (1922; JFM 48.0210.04)] is generalized to a set $$X$$ endowed with finitely many topologies, which are linearly ordered by inclusion. Following S. Plewik and M. Walczyńska [J. Math. 2013, Article ID 289854, 9 p. (2013; Zbl 1284.54006)], the authors estimate the sizes of the composition monoids generated by the operation of complement and the operation of closure for $$n$$ pairwise comparable topologies $$\tau_1\subseteq \tau_2 \subseteq \cdots \subseteq \tau_n$$ on a given set $$X$$. The most important observation states that the considered monoids consist of irreducible words $$x_1\circ x_2 \circ \cdots \circ x_n$$ which are alternating, i.e., every second term is an interior operator and the remaining terms are closure operators. If $$\tau_1\subseteq \tau_2 \subseteq \cdots \subseteq \tau_n$$ is fulfilled, then the word problem is resolvable by the principle: If $$x_1\circ x_3 = x_1$$ and $$x_2\circ x_4 = x_4$$, then $$x_1\circ x_2 \circ x_3 \circ x_4=x_1\circ x_4$$. The particular case of this rule works also for a single topology. Namely, if $$x_1(A) =x_3(A) = \text{interior}(A) = A^{c-c}$$ and $$x_2(A) =x_4(A) = \text{closure}(A) = A^{-}$$, then we get $$A^{c-c-c-c-} = A^{c-c-},$$ which suffices to justify the 14-set closure-complement theorem of Kuratowski.

### MSC:

 54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) 20M20 Semigroups of transformations, relations, partitions, etc.

### Citations:

Zbl 1284.54006; JFM 48.0210.04
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### References:

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