An overview of modern universal algebra

Willard, Ross

doi:10.1017/cbo9780511721151.013

Cited by 4 publications

(4 citation statements)

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“…Basic background in universal algebra for this section can be found in the text by Burris and Sankappanavar [8]. For a convenient and informative summary concentrating on the major developments of the past twenty years we recommend R. Willard's survey [74]. We shall focus on varieties.…”

Section: Applications: Residually Finite Varietiesmentioning

confidence: 99%

“…We have already noted that a residually small and finitely generated congruence modular variety is forced to be residually very finite. Hobby and McKenzie showed that a locally finite variety V satisfying a nontrivial congruence identity is already congruence modular [38, Theorems 9.18 and 10.4] (or see [74,Section 5.7]). This result has since been sharpened to remove the locally finiteness requirement (see Kearnes and Kiss [45, Theorem 9.5]).…”

Section: Applications: Residually Finite Varietiesmentioning

confidence: 99%

“…Taylor's seminal paper [71] gives many conditions equivalent to residual smallness [71] (and see also [74]) We now turn to issues of dualisability. Consider a variety V, not assumed to be of finite type.…”

Section: Residually Smallmentioning

confidence: 99%

“…Semilattice-based algebras have become increasingly prominent, due particularly to their role in R. McKenzie's groundbreaking undecidability results from the late 1990s, following on from [55]; these are discussed in [74]. Semilattices fail to satisfy any nontrivial congruence identity (Freese and Nation [29]).…”

Section: Affine Complete Varieties We Refer To the Monograph Of Kaarmentioning

confidence: 99%

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Natural extensions and profinite completions of algebras

et al. 2011

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This paper investigates profinite completions of residually finite algebras, drawing on ideas from the theory of natural dualities. Given a class A = ISP(M), where M is a set, not necessarily finite, of finite algebras, it is shown that each A ∈ A embeds as a topologically dense subalgebra of a topological algebra n A (A) (its natural extension), and that n A (A) is isomorphic, topologically and algebraically, to the profinite completion of A. In addition it is shown how the natural extension may be concretely described as a certain family of relation-preserving maps; in the special case that M is finite and A possesses a single-sorted or multisorted natural duality, the relations to be preserved can be taken to be those belonging to a dualising set. For an algebra belonging to a finitely generated variety of lattice-based algebras, it is known that the profinite completion coincides with the canonical extension. In this situation the natural extension provides a new concrete realisation of the canonical extension, generalising the well-known representation of the canonical extension of a bounded distributive lattice as the lattice of up-sets of the underlying ordered set of its Priestley dual. The paper concludes with a survey of classes of algebras to which the main theorems do, and do not, apply.

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Section: Applications: Residually Finite Varietiesmentioning

confidence: 99%

Section: Applications: Residually Finite Varietiesmentioning

confidence: 99%