Abstract
The set of equivalence relations on any non-empty set is equipped with a natural order that makes it a complete lattice. The lattice structure only depends on the cardinality of the set, and thus the study of the lattice structure on any countably infinite set is (up to order-isomorphism) the same as studying the lattice of equivalence relations on the set of natural numbers.
We investigate closure under the meet and join operations in the lattice of equivalence relations on the set of natural numbers. Among other results, we show that no set of co-r.e. equivalence relations that contains all logspace-decidable equivalence relations is a lattice.
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Notes
- 1.
Computable reducibility between elements of \(\text {Equ}(\mathbb {N})_{} \) gives rise to an order structure distinct from the classical ordering we consider, and has been investigated in various settings, in particular for \(\varSigma ^0_1\) equivalence relations (called ceers) [1,2,3, 10, 11, 13].
- 2.
A lambda theory is an equivalence relation on the set of closed terms in lambda calculus containing \(\beta \)-equivalence and satisfying a few other natural constraints; see [16] for basic properties.
- 3.
- 4.
Note that \(\mathtt {c}\) does contain the whole tape, of size N and the head position in it. Because the head must point into the tape, the head position is at most N and thus can be stored in binary using only \(\log N\) space. Therefore, counting up to N to actually find the position in the tape requires space \(\log N\) (which is indeed logarithmic in the size of \(\mathtt {c}\)).
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Moyen, JY., Simonsen, J.G. (2021). Subrecursive Equivalence Relations and (non-)Closure Under Lattice Operations. In: De Mol, L., Weiermann, A., Manea, F., Fernández-Duque, D. (eds) Connecting with Computability. CiE 2021. Lecture Notes in Computer Science(), vol 12813. Springer, Cham. https://doi.org/10.1007/978-3-030-80049-9_34
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