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On some unary algebras and their subalgebra lattices. (English) Zbl 1141.08004
For a partial unary algebra $$A$$ let $$S_s(A)$$, $$S_w(A)$$, $$G(A)$$ be the lattice of all strong subalgebras, the lattice of all weak subalgebras, and the digraph of $$A$$, respectively, and similarly for a poset $$(P,\leqq )$$. Further, for a lattice $$L$$, Ir$$(L)$$ be the set of all completely join-irreducible elements, $$D(L)=G($$Ir$$(L),\leqq )$$. The notion of a normality for digraphs, lattices and algebras is introduced. To each normal digraph $$G$$ there is associated a digraph $$TQ(G)$$. It is proved that if $$G$$ is a normal digraph and $$L$$ is a normal lattice, then $$S_s(G)\cong L\iff TQ(G)\cong D(L)$$.
The main result is as follows: Let $$A, B$$ be normal partial unary algebras and $$L$$ be a normal lattice. Then (a) $$S_s(A)\cong L\iff TQ(A)\cong D(L)$$, (b) $$S_s(A)\cong S_s(B)\iff TQ(A)\cong TQ(B)$$.
Applying the above result, necessary and sufficient conditions for lattices $$K, L$$ are found under which there is a normal partial unary algebra $$A$$ such that $$S_w(A)\cong K$$ and $$S_s(A)\cong L$$.

##### MSC:
 08A60 Unary algebras 08A30 Subalgebras, congruence relations 08A55 Partial algebras
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