Extensions and congruences of partial lattices
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Publication:6388364
DOI10.1515/MS-2023-0024arXiv2201.06046MaRDI QIDQ6388364
Publication date: 16 January 2022
Abstract: For a partial lattice L the so-called two-point extension is defined in order to extend L to a lattice. We are motivated by the fact that the one-point extension broadly used for partial algebras does not work in this case, i.e. the one-point extension of a partial lattice need not be a lattice. We describe these two-point extensions and prove several properties of them. We introduce the concept of a congruence on a partial lattice and show its relationship to the notion of a homomorphism and its connections with congruences on the corresponding two-point extension. In particular we prove that the quotient L/E of a partial lattice L by a congruence E on L is again a partial lattice and that the two-point extension of L/E is isomorphic to the quotient lattice of the two-point extension L* of L by the congruence on L* generated by E. Several illustrative examples are enclosed.
Partial orders, general (06A06) Structure theory of lattices (06B05) Lattice ideals, congruence relations (06B10) Partial algebras (08A55) Generalizations of lattices (06B75)
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