A residuated lattice is said to be integrally closed if it satisfies the quasiequations $$xy \le x \implies y \le {\mathrm {e}}$$ x y ≤ x ⇒ y ≤ e and $$yx \le ~x \implies y \le {\mathrm {e}}$$ y x ≤ x ⇒ y ≤ e , or equivalently, the equations $$x \backslash x \approx {\mathrm {e}}$$ x \ x ≈ e and $$x /x \approx {\mathrm {e}}$$ x / x ≈ e . Every integral, cancellative, or divisible residuated lattice is integrally closed, and, conversely, every bounded integrally closed residuated lattice is integral. It is proved that the mapping $$a \mapsto (a \backslash {\mathrm {e}})\backslash {\mathrm {e}}$$ a ↦ ( a \ e ) \ e on any integrally closed residuated lattice is a homomorphism onto a lattice-ordered group. A Glivenko-style property is then established for varieties of integrally closed residuated lattices with respect to varieties of lattice-ordered groups, showing in particular that integrally closed residuated lattices form the largest variety of residuated lattices admitting this property with respect to lattice-ordered groups. The Glivenko property is used to obtain a sequent calculus admitting cut-elimination for the variety of integrally closed residuated lattices and to establish the decidability, indeed PSPACE-completenes, of its equational theory. Finally, these results are related to previous work on (pseudo) BCI-algebras, semi-integral residuated pomonoids, and Casari’s comparative logic.

中文翻译：

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整体封闭的剩余格子

如果剩余格满足拟方程 $$xy \le x \implies y \le {\mathrm {e}}$$ xy ≤ x ⇒ y ≤ e 和 $$yx \le ~x \意味着 y \le {\mathrm {e}}$$ yx ≤ x ⇒ y ≤ e ，或等效地，方程 $$x \backslash x \approx {\mathrm {e}}$$ x \ x ≈ e 和 $ $x /x \approx {\mathrm {e}}$$ x / x ≈ e 。每一个积分的、可取消的或可分的剩余格都是整体闭合的，反之，每一个有界整体闭合的剩余格都是积分的。证明了 $$a \mapsto (a \backslash {\mathrm {e}})\backslash {\mathrm {e}}$$ a ↦ ( a \ e ) \ e 在任何整体闭剩余格上的映射是格序群上的同态。然后，针对各种晶格有序群，为各种整体闭合剩余格建立了 Glivenko 式性质，特别表明，整体闭合剩余格形成了最大种类的剩余格，并承认了关于晶格有序群的这一性质。 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? . Glivenko 性质用于获得一个连续演算，该演算允许对各种整体闭合的剩余格子进行消减，并建立其方程理论的可判定性，实际上是 PSPACE 完备性。最后，这些结果与之前在（伪）BCI 代数、半积分剩余 pomoids 和 Casari 的比较逻辑方面的工作有关。特别表明，整体闭合的剩余格子形成了最大种类的剩余格子，该剩余格子允许关于晶格有序群的这种属性。Glivenko 性质用于获得一个连续演算，该演算允许对各种整体闭合的剩余格子进行消减，并建立其方程理论的可判定性，实际上是 PSPACE 完备性。最后，这些结果与之前在（伪）BCI 代数、半积分剩余 pomoids 和 Casari 的比较逻辑方面的工作有关。特别表明，整体闭合的剩余格子形成了最大种类的剩余格子，该剩余格子允许关于晶格有序群的这种属性。Glivenko 性质用于获得一个连续演算，该演算允许对各种整体闭合的剩余格子进行消减，并建立其方程理论的可判定性，实际上是 PSPACE 完备性。最后，这些结果与之前在（伪）BCI 代数、半积分剩余 pomoids 和 Casari 的比较逻辑方面的工作有关。