圆柱代数

阿尔弗雷德·塔斯基发明的圆柱代数概念自然的出现于一阶逻辑的代数化中。可比较于布尔代数对命题逻辑所扮演的角色。实际上，圆柱代数是装备了建模量化的额外圆柱化运算的布尔代数。

定义

${\displaystyle \alpha }$  维圆柱代数，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 ${\displaystyle (A,+,\cdot ,-,0,1)}$  是布尔代数，${\displaystyle \exists _{i}}$  对于所有 ${\displaystyle i}$  是在 ${\displaystyle A}$  上的一元算子，而对于所有 ${\displaystyle i}$  和 ${\displaystyle j}$ ，${\displaystyle d_{ij}}$ 是${\displaystyle A}$  的指定元素，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 ${\displaystyle \exists _{i}(d_{ij}\cdot x)\cdot \exists _{i}(d_{ij}\cdot -x)=0}$ 

参见

• 抽象代数逻辑
• 一元布尔代数

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.

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