Varieties generated by completions. (English) Zbl 1468.03080
Summary: We prove that persistently finite algebras are not created by completions of algebras, in any ordered discriminator variety. A persistently finite algebra is one without infinite simple extensions. We prove that finite measurable relation algebras are all persistently finite. An application of these theorems is that the variety generated by the completions of representable relation algebras does not contain all relation algebras. This answers Problem 1.1(1) from R. D. Maddux’s 2018 Algebra Universalis paper [Algebra Univers. 79, No. 2, Paper No. 20, 32 p. (2018; Zbl 1444.03177)] in the negative. At the same time, we confirm the suggestion in that paper that the finite maximal relation algebras constructed in M. F. Frias and R. D. Maddux’s 1997 Algebra Universalis paper [Algebra Univers. 38, No. 2, 115–135 (1997; Zbl 0903.03039)] are not in the variety generated by the completions of representable relation algebras. We prove that there are continuum many varieties between the variety generated by the completions of representable relation algebras and the variety of relation algebras.
MSC:
| 03G15 | Cylindric and polyadic algebras; relation algebras |
| 03C05 | Equational classes, universal algebra in model theory |
| 06E25 | Boolean algebras with additional operations (diagonalizable algebras, etc.) |
| 03C13 | Model theory of finite structures |
| 06A06 | Partial orders, general |
| 06B23 | Complete lattices, completions |
| 06A11 | Algebraic aspects of posets |
Keywords:
discriminator varieties; completion; ordered set; dense subalgebra; relation algebra; Boolean algebra with operators; algebraic logic; universal algebraReferences:
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