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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 \(S_s(A)\cong L\).

08A60 Unary algebras
08A30 Subalgebras, congruence relations
08A55 Partial algebras
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