核正则双子代数CRDSA的近布尔性质（三元集划分，CRDSA，嵌入与对偶等价）
The Nearly Boolean Nature of Core Regular Double Sone Algebras, CRDSA (Ternary Set Partitions, CRDSA, Embeddings and Dual Equivalences)
摘要：
在“中心核正则双斯通代数”（CRDSA）中，许多有用的结果表明，开始表明我们在这里集中的CRDSA的近布尔性质。我们定义的节点集格通过众所周知的二元运算的三元集划分，并显示所得到的格同构于C3 ^J，其中C3是3元素链CRDSA。与布尔代数和C2的情形类似，我们得到每个CRDSA都是C3的次直积。我们利用这些结果证明了每个布尔代数都是某个CRDSA的中心。接下来我们证明了C3是原始的，这意味着由它生成的簇对偶等价于布尔代数的范畴。从某种意义上说，这是我们实现将CRDSA建立为接近布尔值的目标的最后一步，但让我们有点不满意。因此，我们继续通过建立一个对偶的范畴之间的CRDSA和专门制作的双拓扑空间。为此目的，我们首先建立充要条件和充分条件对成对零维空间，使它将有一个CRDSA基地B1。我们注意到，这些条件表明CRDSA接近布尔值的程度。例如，如果B1中的u不是开闭/补的，则Cl（u）在B1中并且是开闭/补的。然后我们建立了双连续映射有一个逆是CRDSA同态的充要条件，再次表明CRDSA是布尔的，这些逆必须遵守B1的非闭元边界上的适当条件。在顶点，我们证明了核正则双斯通代数的范畴是对偶等价于我们所说的核正则双成对斯通空间的范畴。我们注意到，这种对偶的条件可以很容易地放松，以产生一个比CRDSA刚性较低的类格的对偶，例如有界分配伪补格。
英文摘要：
In "Centre of Core Regular Double Stone Algebra" (CRDSA), many useful results are shown that begin to indicate the nearly Boolean nature of CRDSA which we focus on here. We define the node set lattice through the well known binary operations of ternary set partitions and show the resultant lattice is isomorphic to C3^J where C3 is the 3 element chain CRDSA. We derive that every CRDSA is a subdirect product of C3 similarly as for Boolean algebras and C2. We use these results to show that every Boolean algebra is the center of some CRDSA. Next we show that C3 is primal implying that the variety generated by it is dually equivalent to the category of Boolean algebras. In some sense this is a last step towards our goal of establishing CRDSA as nearly Boolean, but leaves us a bit dissatisfied. Hence we continue by establishing a duality between the category of CRDSA and specifically crafted bi-topological spaces. Towards this end we first establish necessary and sufficient conditions on a pairwise zero-dimensional space such that it will have a CRDSA base B1. We note that these conditions are indicative of how nearly Boolean CRDSA are. For example, if u in B1 is not clopen/complemented then Cl(u) is in B1 and is clopen/complemented. Then we establish necessary and sufficient conditions for a bi-continuous map to have an inverse that is a CRDSA homomorphism again indicating how nearly Boolean CRDSA are, these inverses must respect the appropriate conditions on the boundary of non-clopen elements of B1. In culmination we show the category of core regular double Stone algebras is dually equivalent to the category of what we call core regular double pairwise Stone spaces. We note that the conditions for this duality can easily be relaxed to yield a duality for a less rigid class lattices than CRDSA, bounded distributive pseudo-complemented lattices for example.
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