In this paper, we employ automated deduction techniques to prove and generalize some well-known theorems in group theory that involve power maps $$ x^n$$ x n . The difficulty lies in the fact that the term $$x^n$$ x n cannot be expressed in the syntax of first-order logic when n is an integer variable. Here we employ a new concept of “power-like functions” by extracting relevant equational properties valid for all power functions and implement these equational rules in Prover9, a first-order theorem prover. We recast the original theorems and prove them in this new context of power-like functions. Consequently these first-order proofs remain valid for all n but the length and complexity of the proofs remain constant independent of the value of n . To give an example, it is well-known (Baer in Proc Am Math Soc 4:15–26, 1953 , Alperin in Can J Math 21:1238–1244 1969 ) that every torsion-free group in which the power map $$f(x) = x^n$$ f ( x ) = x n is an endomorphism is abelian. Here we show that every torsion-free group in which a power-like map is an endomorphism is, indeed, abelian. Also, we generalize similar theorems from groups to a class of cancellative semigroups, and once again, Prover9 happily proves all these new generalizations as well.

中文翻译：

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使用 Power Maps 进行自动推理

在本文中，我们采用自动推导技术来证明和推广群论中一些涉及幂图 $$ x^n$$ xn 的著名定理。困难在于，当 n 是整数变量时，术语 $$x^n$$xn 不能用一阶逻辑的语法表示。在这里，我们通过提取对所有幂函数有效的相关方程属性并在一阶定理证明器 Prover9 中实现这些方程规则，采用了“类幂函数”的新概念。我们改写了原始定理，并在这种类似幂的函数的新上下文中证明了它们。因此，这些一阶证明对所有 n 仍然有效，但证明的长度和复杂性保持不变，与 n 的值无关。举个例子，它是众所周知的（Baer in Proc Am Math Soc 4:15-26, 1953, Alperin 在 Can J Math 21:1238–1244 1969 中）认为幂映射 $$f(x) = x^n$$ f ( x ) = xn 是内同态的每个无扭群都是阿贝尔群。在这里，我们表明，其中幂级映射是内同态的每个无扭群确实是阿贝尔群。此外，我们将群中的类似定理推广到一类取消半群，并且 Prover9 再次愉快地证明了所有这些新的推广。