TY - JOUR

T1 - A Duality for the Variety

AU - Izyurova, A. E.

AU - Schwidefsky, M. V.

N1 - Izyurova, A.E., Schwidefsky, M.V. A Duality for the Variety . Lobachevskii J Math 46, 6125–6142 (2025). https://doi.org/10.1134/S1995080225613864 The research was supported by the Russian Science Foundation, project no. 24-21-00075; https://rscf.ru/project/24-21-00075/.

PY - 2025/12

Y1 - 2025/12

N2 - The diamond is a five-element lattice which is the smallest non-distributive modular lattice. We establish the dual equivalence of the category with objects being bi-algebraic lattices belonging to the variety generated by the diamond and with morphisms being complete lattice homomorphisms and the category with objects being ordered spaces endowed with an equivalence relation and with morphisms preserving the structure of these spaces. A similar result for the pentagon, a five-element lattice which is the smallest non-modular lattice, was established earlier by W. Dziobiak and the second author. The dual spaces here and there are completely different. However, morphisms are, in each case, an application of an old concept—the minimal join cover refinement property for join covers of an element in a lattice.

AB - The diamond is a five-element lattice which is the smallest non-distributive modular lattice. We establish the dual equivalence of the category with objects being bi-algebraic lattices belonging to the variety generated by the diamond and with morphisms being complete lattice homomorphisms and the category with objects being ordered spaces endowed with an equivalence relation and with morphisms preserving the structure of these spaces. A similar result for the pentagon, a five-element lattice which is the smallest non-modular lattice, was established earlier by W. Dziobiak and the second author. The dual spaces here and there are completely different. However, morphisms are, in each case, an application of an old concept—the minimal join cover refinement property for join covers of an element in a lattice.

KW - diamond

KW - duality

KW - lattice

KW - variety

UR - https://www.scopus.com/pages/publications/105036528646

UR - https://www.mendeley.com/catalogue/841027db-9bee-39c7-95bf-1eccca7b3500/

U2 - 10.1134/S1995080225613864

DO - 10.1134/S1995080225613864

M3 - Article

VL - 46

SP - 6125

EP - 6142

JO - Lobachevskii Journal of Mathematics

JF - Lobachevskii Journal of Mathematics

SN - 1995-0802

IS - 12

ER -