可补MacNeille完备化与分式代数
Complemented MacNeille completions and algebras of fractions
摘要：
我们引入（$\ell$ -）双紧性作为半序（格序）集上由两个相容的monoidal结构组成的序代数.双幂零形成了一个适当的框架，研究一般概念的补充，其中包括布尔补有界分配格和乘法逆幺半群。本文的中心问题是双满度是否以及如何嵌入到补双满度中，推广了可消交换幺半群嵌入到其分数群中以及有界分配格嵌入到其自由布尔扩张中。我们证明了每个交换的（$\ell$ -）bimuments确实嵌入到一个完整的可补交换的$\ell$ -bimuments在一个双稠密的方式让人想起Dedekind-MacNeille完成。此外，这个补完备化，它是等价于一个交换对合剩余格的项，有时包含一个更紧的补包络，类似于分数群。在cancellative交换幺半群的情况下，这个代数的分数正是熟悉的一组分数，而在Brouwerian（海廷）代数的情况下，它是一个（有界）幂等对合交换剩余格。这种结构的代数分数实际上产生了一个范畴之间的等价品种的整体和对合剩余结构，其中包括作为特殊情况下，已知的等价之间的阿贝尔$\ell$ -集团和他们的负锥，和杉原幺半群和他们的负锥。
英文摘要：
We introduce ($\ell$-)bimonoids as ordered algebras consisting of two compatible monoidal structures on a partially ordered (lattice-ordered) set. Bimonoids form an appropriate framework for the study of a general notion of complementation, which subsumes both Boolean complements in bounded distributive lattices and multiplicative inverses in monoids. The central question of the paper is whether and how bimonoids can be embedded into complemented bimonoids, generalizing the embedding of cancellative commutative monoids into their groups of fractions and of bounded distributive lattices into their free Boolean extensions. We prove that each commutative ($\ell$-)bimonoid indeed embeds into a complete complemented commutative $\ell$-bimonoid in a doubly dense way reminiscent of the Dedekind--MacNeille completion. Moreover, this complemented completion, which is term equivalent to a commutative involutive residuated lattice, sometimes contains a tighter complemented envelope analogous to the group of fractions. In the case of cancellative commutative monoids this algebra of fractions is precisely the familiar group of fractions, while in the case of Brouwerian (Heyting) algebras it is a (bounded) idempotent involutive commutative residuated lattice. This construction of the algebra of fractions in fact yields a categorical equivalence between varieties of integral and of involutive residuated structures which subsumes as special cases the known equivalences between Abelian $\ell$-groups and their negative cones, and between Sugihara monoids and their negative cones.
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