圓柱代數

阿爾弗雷德·塔斯基發明的圓柱代數概念自然的出現於一階邏輯的代數化中。可比較於布爾代數對命題邏輯所扮演的角色。實際上，圓柱代數是裝備了建模量化的額外圓柱化運算的布爾代數。

定義

${\displaystyle \alpha }$ 維圓柱代數，這裡的 ${\displaystyle \alpha }$ 是任何序數，是代數結構 ${\displaystyle (A,+,\cdot ,-,0,1,\exists _{i},d_{ij})_{i,j<\alpha }}$ 使得 ${\displaystyle (A,+,\cdot ,-,0,1)}$ 是布爾代數，${\displaystyle \exists _{i}}$ 對於所有 ${\displaystyle i}$ 是在 ${\displaystyle A}$ 上的一元算子，而對於所有 ${\displaystyle i}$ 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 ${\displaystyle k\neq i,j}$，則 ${\displaystyle d_{ij}=\exists _{k}(d_{ik}\cdot d_{kj})}$

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

參見

• 抽象代數邏輯
• 一元布爾代數

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.

外部連結

• Jipsen's algebra page.