CREAM：计算$(2^n,1^n)$型代数的[Auto，Endo，Iso，Mono，Epi]-态射、同余、因子等的软件包
CREAM: a Package to Compute [Auto, Endo, Iso, Mono, Epi]-morphisms, Congruences, Divisors and More for Algebras of Type $(2^n,1^n)$
摘要：
CREAM GAP软件包计算具有任意数量的二进制和一元运算的代数的自同构、同余、自同态和子代数;它还决定在两个这样的代数之间是否存在单态、满态、同构或者其中一个是另一个的除数。因此，它发现这些对象几乎所有的代数在实践中使用（群，拟群在其各种签名，半群可能与许多一元运算，领域，半环，quandles，逻辑代数等）。作为一个一刀切的软件包，它只依赖于泛代数定理，而不利用特定的定理，例如，群或半群，以减少搜索空间。Canon和Holt编写了非常快的代码来计算阶数大于128的群的自同构。类似地，Mitchell等人利用深度定理以非常有效的方式计算完全0-单半群的自同构和同余。然而，这些域（128阶以上的群和完全0-单半群）是极少数比我们的通用软件包CREAM更快的GAP代码。对于绝大多数其他类别的代数，要么我们是第一个代码计算上述对象，或现有的算法优于CREAM，在某些情况下，由几个数量级。为了获得这种性能，CREAM使用了泛代数算法与GAP的混合，再加上人工智能定理证明工具（AITP）和非常精细的C实现。作为后者的一个例子，我们重新实现了Freese的非常聪明的算法，用于计算泛代数中的同余，在某种程度上，优于所有其他已知的实现。
英文摘要：
The CREAM GAP package computes automorphisms, congruences, endomorphisms and subalgebras of algebras with an arbitrary number of binary and unary operations; it also decides if between two such algebras there exists a monomorphism, an epimorphism, an isomorphism or if one is a divisor of the other. Thus it finds those objects for almost all algebras used in practice (groups, quasigroups in their various signatures, semigroups possibly with many unary operations, fields, semi-rings, quandles, logic algebras, etc). As a one-size-fits-all package, it only relies on universal algebra theorems, without taking advantage of specific theorems about, eg, groups or semigroups to reduce the search space. Canon and Holt produced very fast code to compute automorphisms of groups that outperform CREAM on orders larger than 128. Similarly, Mitchell et al. take advantage of deep theorems to compute automorphisms and congruences of completely 0-simple semigroups in a very efficient manner. However these domains (groups of order above 128 and completely 0-simple semigroups) are among the very few examples of GAP code faster than our general purpose package CREAM. For the overwhelming majority of other classes of algebras, either ours is the first code computing the above mentioned objects, or the existing algorithms are outperformed by CREAM, in some cases by several orders of magnitude. To get this performance, CREAM uses a mixture of universal algebra algorithms together with GAP coupled with artificial intelligence theorem proving tools (AITP) and very delicate C implementations. As an example of the latter, we re-implement Freese's very clever algorithm for computing congruences in universal algebras, in a way that outperforms all other known implementations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
