The weak subalgebra lattice of a unary partial algebra of a given infinite unary type.

*(English)*Zbl 0986.08004The 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.

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.

Reviewer: Alfonz Haviar (Banská Bystrica)

##### MSC:

08A55 | Partial algebras |

08A60 | Unary algebras |

08A30 | Subalgebras, congruence relations |

05C20 | Directed graphs (digraphs), tournaments |

Full Text:
EuDML

**OpenURL**

##### References:

[1] | BARTOL W.: Weak subalgebra lattices. Comment. Math. Univ. Carolin. 31 (1990), 405-410. · Zbl 0711.08007 |

[2] | BARTOL W.-ROSSELLÓ F.-RUDAK L.: Lectures on Algebras, Equations and Partiality. (F. Rosselló, Technical report B-006, Univ. de les Illes Balears, Dept. Cienc. Mat. Inform., 1992. |

[3] | BERGE C.: Graphs and Hypergraphs. North-Holland, Amsterdam, 1973. · Zbl 0254.05101 |

[4] | BIRKHOFF G. FRINK O.: Representation of lattices by sets. Trans. Amer. Math. Soc. 64 (1948), 299-316. · Zbl 0032.00504 |

[5] | BURMEISTER P.: A Model Theoretic Oriented Approach To Partial Algebras. Math. Res. 32, Akademie Verlag, Berlin, 1986. · Zbl 0598.08004 |

[6] | CRAWLEY P.-DILWORTH R. P.: Algebraic Theory of Lattices. Prentice Hall Inc, Englewood Cliffs, NJ, 1973. · Zbl 0494.06001 |

[7] | GRÄTZER G.: Universal Algebra. (2nd, Springer-Verlag, New York, 1979. · Zbl 0412.08001 |

[8] | JOHNSON J.-SEIFER R. L.: A Survey of Multi-unary Algebras. Mimeographed Seminar Notes, U.C Berkeleu, 1967. |

[9] | JÓNSSON B.: Topics in Universal Algebra. Lecture Notes in Math. 250, Springer-Verlag, New York, 1972. · Zbl 0225.08001 |

[10] | PIÓRO K.: On some non-obvious connections between graphs and unary partial algebras. Czechoslovak Math. J. 50(125) (2000), 295-320. · Zbl 1046.08002 |

[11] | PIÓRO K.: The weak subalgebra lattice of a unary partral algebra of a given finite unary type. Acta Sci. Math. 65 (1999), 439-460. · Zbl 0949.08003 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.