Two remarks on distributivity
Rodolfo C. Ertola-Biraben
UNICAMP

Our first remark concerns the following passage from Peirce in 1880 (see [2, p. 33]): “E. (a + b) × c = (a × c) + (b × c) (a × b) + c = (a + c) × (b + c). These are cases of the distributive principle. They are easily proved by (4) and (2), but the proof is too tedious to give”, where (2) and (4) merely state that × and + behave as the usual infimum and supremum in a lattice. So, Peirce seems to be saying that every lattice is distributive! Now, it is very well known that there are non-distributive lattices. Typical examples are the pentagon and the diamond. So, how can we explain Peirce"s statement? Our second remark regards the notion of distributivity in the case of semilattices. In fact, we will see that there are many such notions. The first one to appear seems to be the one present in [1]. We will see how the different notions relate to each other. Also, we will identify the notion that corresponds to the notion of distributivity in the case of the logic with disjunction as only connective. References [1] Grätzer, G. and Schmidt, E. On congruence lattices of lattices. Acta Math. Acad. Sci. Hungar. 13:179-185, 1962. [2] Peirce, C. S. On the algebra of logic. American Journal of Mathematics 3(1):15-57, 1880.

Lógica
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23/10/2018
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