We investigate a class of combinatory algebras, called ribbon combinatory algebras, in which we can interpret both the braided untyped linear lambda calculus and framed oriented tangles. Any reflexive object in a ribbon category gives rise to a ribbon combinatory algebra. Conversely, From a ribbon combinatory algebra, we can construct a ribbon category with a reflexive object, from which the combinatory algebra can be recovered. To show this, and also to give the equational characterisation of ribbon combinatory algebras, we make use of the internal PRO construction developed in Hasegawa's recent work. Interestingly, we can characterise ribbon combinatory algebras in two different ways: as balanced combinatory algebras with a trace combinator, and as balanced combinatory algebras with duality.

翻译：我们研究一类称为"辫组合代数"的组合代数，其中可以解释无类型线性λ演算与带定向框纽结。任何辫范畴中的自反对象都对应一个辫组合代数。反之，从辫组合代数可构造包含自反对象的辫范畴，且该组合代数能被还原。为证明此结论并建立辫组合代数的等式刻画，我们利用了长谷川近期工作中发展的内部PRO构造。有趣的是，辫组合代数可被两种方式等价刻画：作为带迹算子的平衡组合代数，以及作为带对偶算子的平衡组合代数。