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Distributivity conditions and the order-skeleton of a lattice

Research paper by Jianning Su, Wu Feng, Richard J. Greechie

Indexed on: 30 Oct '11Published on: 30 Oct '11Published in: Algebra universalis



Abstract

We introduce “π-versions” of five familiar conditions for distributivity by applying the various conditions to 3-element antichains only. We prove that they are inequivalent concepts, and characterize them via exclusion systems. A lattice L satisfies D0π if \({{{a}\wedge({b}\vee{c})\;\leqslant\;({a}\wedge{b})\vee{c}}}\) for all 3-element antichains {a, b, c}. We consider a congruence relation ~ whose blocks are the maximal autonomous chains and define the order-skeleton of a lattice L to be \({{\tilde{L} : = L/{\sim}}}\). We prove that the following are equivalent for a lattice L: (i) L satisfies D0π, (ii) \({{\tilde{L}}}\) satisfies any of the five π-versions of distributivity, (iii) the order-skeleton \({{\tilde{L}}}\) is distributive.