Mailbox A new characterization of arithmeticity
Kalle Kaarli · 2001
We prove that an equivalence lattice L EqvA is arithmetical whenever for every 2 L anda 2 A there exists a compatible choice function modulo , havinga as a fixed point. The converse holds if L is finite. For arbitrary set A, all sublattices of EqvA are called equivalence lattices. Let us emphasize that our definition of equivalence lattice does not involve anything else than just being a sublattice of EqvA. In particular, every single equivalence relation ofA forms an equivalence lattice with universe fg. An equivalence lattice L is called arithmetical if it is distributive and every two members of L permute. It is well known that arithmeticity of an equivalence lattice is equivalent to Chinese Remainder Condition (see, for example, (2), Lemma 2.1). Several other characterizations of arithmeticity have been given in terms of compatible functions. A function f on A is said to beL-compatible if it preserves all equivalence relations 2 L. A ternary function p on A is called Pixley function if it satisfies the following equations: p.x;y;y/ Dp.x;y;x/ Dp.y;y;x/ Dx: