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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.

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