10.46298/dmtcs.6904
https://dmtcs.episciences.org/6904
Tóth, Endre
Endre
Tóth
Waldhauser, Tamás
Tamás
Waldhauser
Polymorphism-homogeneity and universal algebraic geometry
We assign a relational structure to any finite algebra in a canonical way,
using solution sets of equations, and we prove that this relational structure
is polymorphism-homogeneous if and only if the algebra itself is
polymorphism-homogeneous. We show that polymorphism-homogeneity is also
equivalent to the property that algebraic sets (i.e., solution sets of systems
of equations) are exactly those sets of tuples that are closed under the
centralizer clone of the algebra. Furthermore, we prove that the aforementioned
properties hold if and only if the algebra is injective in the category of its
finite subpowers. We also consider two additional conditions: a stronger
variant for polymorphism-homogeneity and for injectivity, and we describe
explicitly the finite semilattices, lattices, Abelian groups and monounary
algebras satisfying any one of these three conditions.
Comment: 18 pages, 1 figure
episciences.org
Mathematics - Logic
Mathematics - Rings and Algebras
03C07 (Primary) 03C10, 08A02, 08A35, 08A40, 08B30, 14A99 (Secondary)
arXiv.org - Non-exclusive license to distribute
2021-10-01
2022-03-21
2022-03-21
eng
journal article
arXiv:2007.04405
10.48550/arXiv.2007.04405
1365-8050
https://dmtcs.episciences.org/6904/pdf
VoR
application/pdf
Discrete Mathematics & Theoretical Computer Science
vol. 23 no. 2, special issue in honour of Maurice Pouzet
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