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algebras [2010/08/02 13:25]
jipsen
algebras [2010/08/02 13:42] (current)
jipsen
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  *[[3-element cyclic group]] $\mathbb Z_3=\langle\{0,1,2\},+_3,-_3,0\rangle$   *[[3-element cyclic group]] $\mathbb Z_3=\langle\{0,1,2\},+_3,-_3,0\rangle$
  *[[3-element field]] $\mathbb F_3=\langle\{0,1,2\},+_3,-_3,0,\cdot_3,1\rangle$   *[[3-element field]] $\mathbb F_3=\langle\{0,1,2\},+_3,-_3,0,\cdot_3,1\rangle$
-  *[[3-element semilattices]] $\mathbb S_{3,0}=\langle\{0,1,2\},\cdot\rangle$, $\mathbb S_{3,1}=\langle\{0,1,2\},\cdot\rangle$+  *[[3-element semilattices]] $\mathbb S_{3,0}=\langle\{0,1,2\},\cdot\rangle$, $\mathbb S_{3,1}=\langle\{0,1,2\},\min\rangle$
====Some 4-element algebras==== ====Some 4-element algebras====
-  *[[4-element boolean algebra]] $(\mathbb B_2)^2=\langle\{0,1,2,3\},\vee,0,\wedge,3,'\rangle$+  *[[4-element boolean algebra]] $\mathbb B_2^2=\langle\{0,1,2,3\},\vee,0,\wedge,3,'\rangle$
  *[[4-element chain]] $\mathbb C_4=\langle\{0,1,2,3\},\vee,\wedge\rangle$   *[[4-element chain]] $\mathbb C_4=\langle\{0,1,2,3\},\vee,\wedge\rangle$
  *[[4-element cyclic group]] $\mathbb Z_4=\langle\{0,1,2,3\},+_4,-_4,0\rangle$   *[[4-element cyclic group]] $\mathbb Z_4=\langle\{0,1,2,3\},+_4,-_4,0\rangle$
-  *[[4-element distributive lattice]] $(\mathbb C_2)^2=\langle\{0,1,2,3\},\vee,\wedge\rangle$ +  *[[4-element distributive lattice]] $(\mathbb C_2)^2=\langle\{(0,0),(0,1),(1,0),(1,1)\},\vee,\wedge\rangle$ 
-  *[[4-element field]] $\mathbb F_4=\langle\{0,1,x,x+1\},+_2,-_2,0,\cdot,1\rangle=F[x]/\langle x^2+x+1\rangle$+  *[[4-element field]] $\mathbb F_4=\langle\{0,1,x,x+1\},+_2,-_2,0,\cdot,1\rangle\cong\mathbb F_2[x]/\langle x^2+x+1\rangle$
  *[[4-element noncyclic group]] $(\mathbb Z_2)^2=\langle\{(0,0),(0,1),(1,0),(1,1)\},+_2,-_2,(0,0)\rangle$   *[[4-element noncyclic group]] $(\mathbb Z_2)^2=\langle\{(0,0),(0,1),(1,0),(1,1)\},+_2,-_2,(0,0)\rangle$
 +  *[[4-element nonunital rings]] $\mathbb Z_{4,0}=\langle\{0,1,2,3\},+_4,-_4,0,\cdot_0\rangle$, $(\mathbb F_2)^2=\langle\{(0,0),(0,1),(1,0),(1,1)\},+_2,-_2,(0,0),\cdot_0\rangle$
 +  *[[4-element unital rings]] $\mathbb Z_{4,1}=\langle\{0,1,2,3\},+_4,-_4,0,\cdot_4,1\rangle$, $(\mathbb F_2)^2=\langle\{(0,0),(0,1),(1,0),(1,1)\},+_2,-_2,(0,0),\cdot,(1,1)\rangle$
====Some 5-element algebras==== ====Some 5-element algebras====
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====Some n-element algebras==== ====Some n-element algebras====
-  *[[2^k-element boolean algebra]] $(\mathbb B_2)^k=\langle\{0,1,2,\ldots,2^k-1\},\vee,0,\wedge,2^k-1,'\rangle$+  *[[2^k-element boolean algebra]] $\mathbb B_2^k=\langle\{0,1,2,\ldots,2^k-1\},\vee,0,\wedge,2^k-1,'\rangle$ 
 +  *[[k!-element symmetric group]] $S_k=\langle\{$permutations on $k$-element set$\},\circ,{}^{-1},()\rangle$
  *[[n-element chain]] $\mathbb C_n=\langle\{0,1,2,\ldots,n-1\},\vee,\wedge\rangle$   *[[n-element chain]] $\mathbb C_n=\langle\{0,1,2,\ldots,n-1\},\vee,\wedge\rangle$
  *[[n-element cyclic group]] $\mathbb Z_n=\langle\{0,1,2,\ldots,n-1\},+_n,-_n,0\rangle$   *[[n-element cyclic group]] $\mathbb Z_n=\langle\{0,1,2,\ldots,n-1\},+_n,-_n,0\rangle$
-  *[[p^k-element field]] $\mathbb F_{p^k}=\langle\{0,1,\ldots,p-1,x,\ldots\},+_p,-_p,0,\cdot,1\rangle=F[x]/\langle p(x)\rangle$+  *[[p^k-element field]] $\mathbb F_{p^k}=\langle\{0,1,\ldots,p-1,x,\ldots\},+_p,-_p,0,\cdot,1\rangle\cong\mathbb F_p[x]/\langle f(x)\rangle$
====Some infinite algebras==== ====Some infinite algebras====