First we identify the free algebras of the class of algebras of binary relations equipped with the composition and domain operations. Elements of the free algebras are pointed labelled finite rooted trees. Then we extend to the analogous case when the signature includes all the Kleene algebra with domain operations; that is, we add union and reflexive transitive closure to the signature. In this second case, elements of the free algebras are ‘regular’ sets of the trees of the first case. As a corollary, the axioms of domain semirings provide a finite quasiequational axiomatisation of the equational theory of algebras of binary relations for the intermediate signature of composition, union, and domain. Next we note that our regular sets of trees are not closed under complement, but prove that they are closed under intersection. Finally, we prove that under relational semantics the equational validities of Kleene algebras with domain form a decidable set.

首先，我们确定具有组合和域运算的二元关系代数类的自由代数。自由代数的元素是尖的，标记为有限根树。然后，当签名包括具有域运算的所有Kleene代数时，我们扩展到类似情况。也就是说，我们在签名中添加并集和自反传递封闭。在第二种情况下，自由代数的元素是第一种情况的“正规”树集。作为推论，领域半环的公理为二元关系代数方程组理论的有限quasquaquational公理化提供了组成，联合和领域的中间特征。接下来，我们注意到我们的常规树不是在补数下闭合，而是证明它们在交点下闭合。最后，