Generalized bunched implication algebras (GBI-algebras) are defined as residuated lattices with a Heyting implication, and are positioned between Boolean algebras with operators and lattices with operators. We characterize congruences on GBI-algebras by filters that are closed under Gumm–Ursini terms, and for involutive GBI-algebras these terms simplify to a dual version of the congruence term for relation algebras together with two more terms. We prove that representable weakening relation algebras form a variety of cyclic involutive GBI-algebras, denoted by RWkRA, containing the variety of representable relation algebras. We describe a double-division conucleus construction on residuated lattices and on (cyclic involutive) GBI-algebras and show that it generalizes Comer’s double coset construction for relation algebras. Also, we explore how the double-division conucleus construction interacts with other class operators and in particular with variety generation. We focus on the fact that it preserves a special discriminator term, thus yielding interesting discriminator varieties of GBI-algebras, including RWkRA. To illustrate the generality of the variety of weakening relation algebras, we prove that all distributive lattice-ordered pregroups and hence all lattice-ordered groups embed, as residuated lattices, into representable weakening relation algebras on chains. Moreover, every representable weakening relation algebra is embedded in the algebra of all residuated maps on a doubly-algebraic distributive lattice. We give a number of other instructive examples that show how the double-division conucleus illuminates the structure of distributive involutive residuated lattices and GBI-algebras.

广义束含蕴涵代数（GBI-algebras）定义为具有Heyting蕴涵的剩余格，并位于带算子的布尔代数和带算子的晶格之间。我们用在Gumm–Ursini项下封闭的过滤器来表征GBI代数的同余，对于渐进式GBI代数，这些项简化为关系代数的同余项的对偶形式以及另外两个。我们证明了可表示的弱关系代数形成了各种循环对合GBI代数，用RWkRA表示，其中包含各种可表示的关系代数。我们在剩余格上和（循环渐开线）GBI代数上描述了一个双分割的conucleus构造，并表明它推广了关系代数的Comer的双陪集构造。此外，我们探讨了双分体的构造如何与其他类别的运算符，尤其是与品种生成相互作用。我们关注的事实是它保留了一个特殊的鉴别词，从而产生了有趣的GBI代数鉴别词，包括RWkRA。为了说明各种弱化关系代数的一般性，我们证明了所有分布的有序有序的预群以及所有有序的群作为残差格嵌入到链上可表示的弱化关系代数中。此外，每个可表示的弱化关系代数都嵌入在双代数分布格上所有剩余映射的代数中。我们还提供了许多其他说明性示例，这些示例说明了双分体的凹线形如何阐明分布式渐开线剩余格和GBI代数的结构。