All minimal clones on the three-element set
All minimal clones on the three-element set
复制标题
三元素集上的所有最小克隆
作者：
B. Csákány
集合M上的克隆是M上的酉运算的集合，它在合成下是封闭的，并且包含所有的投影。M上的克隆构成一个代数格，该格上的原子和对偶原子分别称为M上的极小克隆和极大克隆。Post给出了所有克隆的完整描述，因此给出了M2的所有最小克隆和最大克隆的完整描述；对于M|=3，Jablonskil找到了所有最大克隆的完整列表，而Rosenberg对于任何有限M(见[15]、[10]和[17])找到了所有最大克隆的完整列表。到目前为止，对于案例|M|>；2，只有最小克隆的特殊例子是已知的。在本文中，我们确定了三元M上的所有极小克隆。我们使用标准的通用代数术语[9]，只是函数代表运算，而术语函数代表多项式。所有函数(因此所有克隆)都定义在基本集合3={0，1，2}上。如果/是一个函数，则[/]是由/生成的克隆，即代数(3；/)的所有项函数的克隆。投影也称为平凡函数。我们使用符号a表示由来自3的不同项组成的三元组集合，i表示3\<；r。在下文中，我们经常使用以下类型的函数1)-4)。1)一元函数。这样的函数f由u“表示，其中？=9./(0)4+3./(L)+/(2)。2)二元幂等函数。Cayley表中的这样一个函数
A clone on a set M is a set of Unitary operations on M which is closed under composition and contains all projections. The clones on M form an algebraic lattice; the atoms and the dual atoms of this lattice are called minimal clones and maximal clones on M, respectively. A full description of all clones, hence of all minimal and maximal clones for \M\ = 2 was given by Post; a complete list of all maximal clones was found by Jablonskil for \M | = 3 and by Rosenberg for any finite M (see [15], [10], and [17]). Until now, only special examples of minimal clones were known for the case |M|>2 . In this paper we determine all minimal clones on a three-element M. We use the standard universal algebraic terminology [9] except that function stands for operation and term function for polynomial. All functions (and hence all clones) are defined on the base set 3 = {0, 1, 2}. If / is a function, [ / ] is the clone generated by / i.e. the clone of all term functions of the algebra (3; / ) . Projections will also be called trivial functions. We use the notation a for the set of triplets consisting of distinct entries from 3 and i for 3\<r. In what follows we often make use of functions of the following types 1)—4). 1) Unary functions. Such a function / is denoted by u„, where «=9./(0)4+ 3. /( l )+/(2) . 2) Binary idempotent functions. Such a function with the Cayley table