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Dominions in quasivarieties of universal algebras

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Abstract

The dominion of a subalgebra H in an universal algebra A (in a class \(\mathcal{M}\)) is the set of all elements \(a \in A\) such that for all homomorphisms \(f,g:A \to B \in \mathcal{M}\) if f, g coincide on H, then af = ag. We investigate the connection between dominions and quasivarieties. We show that if a class \(\mathcal{M}\) is closed under ultraproducts, then the dominion in \(\mathcal{M}\) is equal to the dominion in a quasivariety generated by \(\mathcal{M}\). Also we find conditions when dominions in a universal algebra form a lattice and study this lattice.

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Special issue of Studia Logica: “Algebraic Theory of Quasivarieties” Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko

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Budkin, A. Dominions in quasivarieties of universal algebras. Stud Logica 78, 107–127 (2004). https://doi.org/10.1007/s11225-005-7127-1

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Keywords

  • Quasivariety
  • dominion
  • universal algebra
  • group
  • lattice
  • free amalgamated product
  • amalgam