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The weak subalgebra lattice of a unary partial algebra of a given infinite unary type. (English) Zbl 0986.08004
The aim of the paper is to characterize the weak subalgebra lattice of a unary partial algebra of a given infinite unary type. W. Bartol proved that a lattice \(\mathcal L\) is isomorphic to a weak subalgebra lattice \(S_w(\mathcal A)\) (for some partial algebra \(\mathcal A\)) iff \(\mathcal L\) is algebraic and distributive, every element is a join of join irreducible elements, every set \(\mathcal At(i)\) of all atoms \(a\) with \(a \leq i\) is finite (and nonempty), for any non-zero join-irreducible element \(i\) with \(1 \leq |\mathcal A(i)|\leq 2\) if the algebra \(\mathcal A\) is unary, and the set \(\mathcal Ir(\mathcal L)\) of all non-zero and non-atomic join-irreducible elements is an antichain of \(\mathcal L\).
The main result of the present paper is the following theorem.
Theorem. Let \(K\) be an infinite unary algebraic type and let \(\mathcal L\) be a lattice which satisfies Bartol’s conditions. Then the following conditions are equivalent:
(a) There is a unary partial algebra \(\mathcal A\) of the type \(K\) such that its weak
subalgebra lattice \(S_w(\mathcal A)\) is isomorphic to \(\mathcal L\).
(b) \(\mathcal L\) satisfies the following conditions:
(b1) \(|\{i\in \mathcal Ir(\mathcal L): \mathcal At(i)=\{a,b\}\}|\leq |K|\), for any atoms \(a\), \(b\),
(b2) there exists an algebraic closure operator \(C_{\mathcal L}\) on the set \(\mathcal At(\mathcal L)\) of all atoms of \(\mathcal L\) such that for every \(B\subseteq \mathcal At(\mathcal L)\) \(|C_{\mathcal L}(B)|\leq \max \{|K|_1,|B|\}\), \(|\{ b\in C_{\mathcal L}(B): (\exists i \in \mathcal Ir(\mathcal L)) (\mathcal At(i)=\{a,b\})\}|\leq |K|\) for each \(a \in \mathcal A(L)-C_{\mathcal L}(B)\), where \(\mathcal Ir(\mathcal L)\) is the set of all non-zero and non-atomic join-irreducible elements, \(\mathcal At(i)\) is the set of all atoms \(a\) of \(\mathcal L\) with \(a\leq i\) and \(|K|_1\) is the least cardinal number greater than \(|K|\).
The result is obtained by using immediate connections between unary partial algebras and digraphs (from K. Pióro [“On some non-obvious connections between graphs and unary partial algebras”, Czech. Math. J. 50, 295-320 (2000)]) and by using transfinite induction.

08A55 Partial algebras
08A60 Unary algebras
08A30 Subalgebras, congruence relations
05C20 Directed graphs (digraphs), tournaments
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