We investigate a number of semantically defined fragments of Tarski's algebra of binary relations, including the function-preserving fragment. We address the question whether they are generated by a finite set of operations. We obtain several positive and negative results along these lines. Specifically, the homomorphism-safe fragment is finitely generated (both over finite and over arbitrary structures). The function-preserving fragment is not finitely generated (and, in fact, not expressible by any finite set of guarded second-order definable function-preserving operations). Similarly, the total-function-preserving fragment is not finitely generated (and, in fact, not expressible by any finite set of guarded second-order definable total-function-preserving operations). In contrast, the forward-looking function-preserving fragment is finitely generated by composition, intersection, antidomain, and preferential union. Similarly, the forward-and-backward-looking injective-function-preserving fragment is finitely generated by composition, intersection, antidomain, inverse, and an `injective union' operation.

翻译：我们研究了塔尔斯基二元关系代数中若干语义定义的片段，包括函数保持片段。我们探讨了它们是否可由有限运算集生成的问题。沿着这一思路，我们获得了若干肯定与否定结果。具体而言，同态安全片段是有限生成的（在有限结构及任意结构上均成立）。函数保持片段不是有限生成的（事实上，无法用任何有限集的有界二阶可定义函数保持运算表示）。类似地，全函数保持片段也不是有限生成的（事实上，无法用任何有限集的有界二阶可定义全函数保持运算表示）。与之相对，前向函数保持片段可通过复合、交、反定义域及优先并运算有限生成。类似地，前向与后向单射函数保持片段可通过复合、交、反定义域、逆及“单射并”运算有限生成。