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**Subdirectly irreducible algebras of quasiordered logics.**
*(English)*
Zbl 0799.06026

A \(q\)-lattice is an algebra \({\mathcal A}= (A,\vee, \wedge)\) with two binary operations which are associative and commutative and satisfy the following axioms: \(x\vee (x\wedge y)= x\vee x\) and the dual, \(x\vee (y\vee y)= x\vee y\) and the dual, \(x\vee x= x\wedge x\). Distributivity of a \(q\)-lattice is defined by the identity \(x\wedge (y\vee z)= (x\wedge y)\vee (x\wedge z)\). A \(q\)-lattice \({\mathcal A}\) is called bounded if there exist element 0 and 1 of \(A\) such that \(0\wedge x=0\), \(1\vee x=1\) for all \(x,y\in A\). A bounded \(q\)-lattice \({\mathcal A}\) is said to be complementary if for each \(a\in A\) there exists \(b\in A\) (a complement of \(a\)) with \(a\vee b=1\), \(a\wedge b=0\). Note that if \(b\), \(c\) are complements of \(a\), then \(b\vee b=c\vee c\). An algebra \((A,\vee, \wedge,',0,1)\) of type \((2,2,1,0,0)\) is called an algebra of quasiordered logic if \((A,\vee, \wedge, 0,1)\) is a bounded distributive \(q\)-lattice and \('\) is defined as \(a\mapsto a'= b\vee b\), where \(b\) is a complement of \(a\). It is proved that an algebra of quasiordered logic is subdirectly irreducible iff it has either two or three elements (there exists three such algebras).

Reviewer: V.N.Salij (Saratov)

### MSC:

06E99 | Boolean algebras (Boolean rings) |

08A05 | Structure theory of algebraic structures |

08B26 | Subdirect products and subdirect irreducibility |

06E05 | Structure theory of Boolean algebras |

### Keywords:

quasiorder; Boolean algebra; subdirect irreducibility; \(q\)-lattice; algebra of quasiordered logic### References:

[1] | Birkhoff G.: Subdirect unions in universal algebras. Bull. Amer. Math. Soc. 50 (1944), 764-768. · Zbl 0060.05809 · doi:10.1090/S0002-9904-1944-08235-9 |

[2] | Chajda I.: Lattices in quasiordered sets. Acta UPO, Fac. Rer. Nat., Math. XXXI, Vol. 105, 31(1992), 6-12. · Zbl 0773.06002 |

[3] | Chajda I.: An algebra of quasiordered logic. Math. Bohemica, to appear. · Zbl 0816.06007 |

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