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Ultralocally Closed Clones
Keith A. Kearnes and Ágnes Szendrei
Let 𝐴 and 𝐼 be sets. For every 𝓃-ary operation 𝑓 : 𝐴𝓃 → 𝐴, and for every ultrafilter 𝒰 on 𝐼, 𝑓 has an extension 𝑓𝒰 to the ultrapower A𝓊 := A𝐼 /𝒰, defined as follows:
𝑓𝓊 (𝑎1/𝒰, . . . , 𝑎𝑛/𝒰) = 𝑓(𝑎1, . . . , 𝑎𝑛)/𝒰 for 𝑎𝑖 ∈ 𝐴𝐼 .
For any clone 𝖢 on 𝐴 and ultrafilter 𝒰 on 𝐼, the clone 𝖢 extends to a clone 𝖢𝓊 := {𝑔𝓊 : g ∈ 𝖢} on 𝐴𝓊. We define the ultralocal closure of C to be the clone of all operations 𝑓 : 𝐴𝓃 → 𝐴 such that 𝑓𝓊 is in the local closure of 𝖢𝓊 for all choices of 𝐼 and 𝒰.
Our main result is a characterization of the ultralocal closure of a clone in terms of an interpolability property. We prove an “ultra” version of the Baker–Pixley Theorem, and we show that the clone of any simple module is ultralocally closed.
Keywords: Baker-Pixley Theorem, clone, interpolation, local closure, local operation, ultralocal closure, ultrapower
2020 Mathematics Subject Classification. Primary: 08A40; Secondary: 03C20