AI 中文总结
该研究针对强剩余有限代数，证明这类代数上函数保持同余/稳定预序，与保持可识别集生成的格/布尔代数原像封闭这两种性质等价。
AI 中文摘要
我们考察一些幺半群和（半）环（自然数、整数与p进整数），更一般地，考察强意义下的剩余有限代数，证明这类代数上的函数f表现得如同代数运算的两种方式等价：第一种是保持同余或稳定预序；第二种是要求对任意可识别集L，由L生成的合适格（或布尔代数）在f的原像下封闭。
英文摘要
Looking at some monoids and (semi)rings (natural numbers, integers and $p$-adic integers), and more generally, residually finite algebras (in a strong sense), we prove the equivalence of two ways for a function $f$ on such an algebra to behave like the operations of the algebra. The first way is to preserve congruences or stable preorders. The second way is to demand that, for any (recognizable) set $L$, a suitably chosen lattice (or Boolean algebra) generated by $L$ be closed under inverse images by the function $f$.
