A modular semilattice is a semilattice generalization of a modular lattice. We establish a Birkhoff-type representation theorem for modular semilattices, which says that every modular semilattice is isomorphic to the family of ideals in a certain poset with additional relations.This new poset structure, which we axiomatize in this paper, is called a PPIP (projective poset with inconsistent pairs). A PPIP is a common generalization of a PIP (poset with inconsistent pairs) and a projective ordered space. The former was introduced by Barth\'elemy and Constantin for establishing Birkhoff-type theorem for median semilattices, and the latter by Herrmann, Pickering, and Roddy for modular lattices. We show the $\Theta (n)$ representation complexityand a construction algorithm for PPIP-representations of $(\wedge, \vee)$-closed sets in the product $L^n$ of modular semilattice $L$. This generalizes the results of Hirai and Oki for a special median semilattice $S_k$. We also investigate implicational bases for modular semilattices. Extending earlier results of Wild and Herrmann for modular lattices, we determine optimal implicational bases and develop a polynomial time recognition algorithm for modular semilattices. These results can be applied to retain the minimizer set of a submodular function on a modular semilattice.

中文翻译：

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模半格的紧凑表示及其应用

模半格是模格的半格推广。我们建立了模半格的 Birkhoff 型表示定理，它说每个模半格都同构于具有附加关系的某个偏序组中的理想族。我们在本文中公理化的这种新的偏序组结构称为 PPIP（具有不一致对的投影poset）。PPIP 是 PIP（具有不一致对的poset）和投影有序空间的常见概括。前者是 Barth\'elemy 和 Constantin 为建立中值半格的 Birkhoff 型定理而引入的，后者是由 Herrmann、Pickering 和 Roddy 引入的用于模格的。我们展示了 $\Theta (n)$ 表示复杂度和 $(\wedge, \vee)$-模半格 $L$ 的积 $L^n$ 中的闭集。这概括了 Hirai 和 Oki 对特殊中值半格 $S_k$ 的结果。我们还研究了模半格的蕴涵基。扩展 Wild 和 Herrmann 对模格的早期结果，我们确定了最佳蕴涵基并开发了一种用于模半格的多项式时间识别算法。这些结果可用于保留模半格上的子模函数的最小集合。我们确定了最佳蕴涵基并开发了一种用于模半格的多项式时间识别算法。这些结果可用于保留模半格上的子模函数的最小集合。我们确定了最佳蕴涵基并开发了一种用于模半格的多项式时间识别算法。这些结果可用于保留模半格上的子模函数的最小集合。