# 圆柱代数

## 定义

${\displaystyle \alpha }$ 圆柱代数，这里的 ${\displaystyle \alpha }$  是任何序数，是代数结构 ${\displaystyle (A,+,\cdot ,-,0,1,\exists _{i},d_{ij})_{i,j<\alpha }}$  使得 ${\displaystyle (A,+,\cdot ,-,0,1)}$ 布尔代数${\displaystyle \exists _{i}}$  对于所有 ${\displaystyle i}$  是在 ${\displaystyle A}$  上的一元算子，而 ${\displaystyle d_{ij}}$  对于所有 ${\displaystyle i}$ ${\displaystyle j}$  是在 ${\displaystyle A}$  上的一个显著的元素，使得如下成立:

(C1) ${\displaystyle \exists _{i}0=0\,}$

(C2) ${\displaystyle x\leq \exists _{i}x}$

(C3) ${\displaystyle \exists _{i}(x\cdot \exists _{i}y)=\exists _{i}x\cdot \exists _{i}y}$

(C4) ${\displaystyle \exists _{i}\exists _{j}x=\exists _{j}\exists _{i}x\,}$

(C5) ${\displaystyle d_{ii}=1\,}$

(C6) 如果 ${\displaystyle k\neq i,j}$ ，则 ${\displaystyle d_{ij}=\exists _{k}(d_{ik}\cdot d_{kj})}$

(C7) 如果 ${\displaystyle i\neq j}$ ，则 ${\displaystyle \exists _{i}(d_{ij}\cdot x)\cdot \exists _{i}(d_{ij}\cdot -x)=0}$

## 引用

• Leon Henkin, Monk, J.D., and Alfred Tarski (1971) Cylindric Algebras, Part I. North-Holland. ISBN 978-0-7204-2043-2.
• -------- (1985) Cylindric Algebras, Part II. North-Holland.
• Caleiro, C., and Gonçalves, R (2007) "On the algebraization of many-sorted logics" in J. Fiadeiro and P.-Y. Schobbens, eds., Recent Trends in Algebraic Development Techniques - Selected Papers, Vol. 4409 of Lecture Notes in Computer Science. Springer-Verlag: 21-36.