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algebras [2010/08/01 18:20]
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algebras [2010/08/02 13:42] (current)
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=====Algebras===== =====Algebras=====
 +
 +====Some 1-element algebras====
 +
 +  *[[1-element boolean algebra]] $\mathbb B_1=\langle\{0\},\vee,0,\wedge,0,'\rangle$
 +  *[[1-element chain]] $\mathbb C_1=\langle\{0\},\vee,\wedge\rangle$
 +  *[[1-element group]] $\mathbb Z_1=\langle\{0\},+_1,-_1,0\rangle$
 +  *[[1-element mono-unary algebra]] $\mathbb U_1=\langle\{0\},0\rangle$
 +  *[[1-element semilattice]] $\mathbb S_1=\langle\{0\},\cdot\rangle$
 +  *[[1-element set]] $\mathbf 1=\langle\{0\}\rangle$
====Some 2-element algebras==== ====Some 2-element algebras====
-  *[[2-element abelian group]] +  *[[2-element boolean algebra]] $\mathbb B_2=\langle\{0,1\},\vee,0,\wedge,1,'\rangle$ 
-  *[[2-element boolean algebra]] +  *[[2-element chain]] $\mathbb C_2=\langle\{0,1\},\vee,\wedge\rangle$ 
-  *[[2-element distributive lattice]] +  *[[2-element cyclic group]] $\mathbb Z_2=\langle\{0,1\},+_2,-_2,0\rangle$ 
-  *[[2-element field]] +  *[[2-element field]] $\mathbb F_2=\langle\{0,1\},+_2,-_2,0,\cdot_2,1\rangle$ 
-  *[[2-element mono-unary algebras]] +  *[[2-element mono-unary algebras]] $\mathbb U_{2,0}=\langle\{0,1\},0\rangle$, $\mathbb U_{2,1}=\langle\{0,1\},1\rangle$, $\mathbb U_{2,'}=\langle\{0,1\},'\rangle$ 
-  *[[2-element semilattice]]+  *[[2-element semilattice]] $\mathbb S_2=\langle\{0,1\},\cdot\rangle$ 
 +  *[[2-element set]] $\mathbf 2=\langle\{0,1\}\rangle$
====Some 3-element algebras==== ====Some 3-element algebras====
-  *[[3-element abelian group]] +  *[[3-element chain]] $\mathbb C_3=\langle\{0,1,2\},\vee,\wedge\rangle$ 
-  *[[3-element distributive lattice]] +  *[[3-element cyclic group]] $\mathbb Z_3=\langle\{0,1,2\},+_3,-_3,0\rangle$ 
-  *[[3-element field]] +  *[[3-element field]] $\mathbb F_3=\langle\{0,1,2\},+_3,-_3,0,\cdot_3,1\rangle$ 
-  *[[3-element mono-unary algebras]] +  *[[3-element semilattices]] $\mathbb S_{3,0}=\langle\{0,1,2\},\cdot\rangle$, $\mathbb S_{3,1}=\langle\{0,1,2\},\min\rangle$ 
-  *[[3-element semilattices]]+
====Some 4-element algebras==== ====Some 4-element algebras====
-  *[[4-element abelian groups]] +  *[[4-element boolean algebra]] $\mathbb B_2^2=\langle\{0,1,2,3\},\vee,0,\wedge,3,'\rangle$ 
-  *[[4-element boolean algebra]] +  *[[4-element chain]] $\mathbb C_4=\langle\{0,1,2,3\},\vee,\wedge\rangle$ 
-  *[[4-element distributive lattices]] +  *[[4-element cyclic group]] $\mathbb Z_4=\langle\{0,1,2,3\},+_4,-_4,0\rangle$ 
-  *[[4-element field]] +  *[[4-element distributive lattice]] $(\mathbb C_2)^2=\langle\{(0,0),(0,1),(1,0),(1,1)\},\vee,\wedge\rangle$ 
-  *[[4-element semilattices]]+  *[[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 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====
-  *[[5-element abelian group]] +  *[[5-element chain]] $\mathbb C_5=\langle\{0,1,2,3,4\},\vee,\wedge\rangle$ 
-  *[[5-element distributive lattices]] +  *[[5-element cyclic group]] $\mathbb Z_5=\langle\{0,1,2,3,4\},+_5,-_5,0\rangle$ 
-  *[[5-element semilattices]]+  /*[[5-element distributive lattices]]*/
====Some 6-element algebras==== ====Some 6-element algebras====
-  *[[6-element nonabelian group]]+  *[[6-element nonabelian group]] $S_3=\langle\{(),(12),(13),(23),(123),(132)\},\circ,{}^{-1},()\rangle$ 
 + 
 +====Some n-element algebras==== 
 + 
 +  *[[2^k-element boolean algebra]] $\mathbb B_2^k=\langle\{0,1,2,\ldots,2^k-1\},\vee,0,\wedge,2^k-1,'\rangle$ 
 +  *[[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 cyclic group]] $\mathbb Z_n=\langle\{0,1,2,\ldots,n-1\},+_n,-_n,0\rangle$ 
 +  *[[p^k-element field]] $\mathbb F_{p^k}=\langle\{0,1,\ldots,p-1,x,\ldots\},+_p,-_p,0,\cdot,1\rangle\cong\mathbb F_p[x]/\langle f(x)\rangle$
====Some infinite algebras==== ====Some infinite algebras====
-  *[[Positive integers additive semigroup]] +  *[[Positive integers additive semigroup]] $\langle \mathbb Z^+,+\rangle$ 
-  *[[Positive integers monoid]] +  *[[Positive integers monoid]] $\langle \mathbb Z^+,\cdot,1\rangle$ 
-  *[[Natural numbers additive monoid]] +  *[[Natural numbers additive monoid]] $\langle \mathbb N,+,0\rangle$ 
-  *[[Natural numbers multiplicative monoid]] +  *[[Natural numbers multiplicative monoid]] $\langle \mathbb N,\cdot,1\rangle$ 
-  *[[Integers additive group]] +  *[[Integers additive group]] $\langle \mathbb Z,+,-,0\rangle$ 
-  *[[Integers ring]] +  *[[Integers ring]] $\langle \mathbb Z,+,-,0,\cdot,1\rangle$ 
-  *[[Integers lattice-ordered group]] +  *[[Integers lattice-ordered group]] $\langle \mathbb Z,\vee,\wedge,+,-,0\rangle$ 
-  *[[Rational numbers additive group]] +  *[[Rational numbers additive group]] $\langle \mathbb Q,+,-,0\rangle$ 
-  *[[Positive rational numbers group]] +  *[[Positive rational numbers group]] $\langle \mathbb Q^+,\cdot,{}^{-1},1\rangle$ 
-  *[[Rational numbers field]] +  *[[Rational numbers field]] $\langle \mathbb Q,+,-,0,\cdot,1\rangle$ 
-  *[[Real numbers field]] +  *[[Real numbers field]] $\langle \mathbb R,+,-,0,\cdot,1\rangle$ 
-  *[[Gaussian integers ring]] +  *[[Gaussian integers ring]] $\langle \mathbb Z[i],+,-,0,\cdot,1\rangle$ 
-  *[[Complex numbers field]]+  *[[Complex numbers field]] $\langle \mathbb C,+,-,0,\cdot,1\rangle$
-  *[[Natural number lattice]] +  *[[Natural number chain]] $\langle \mathbb N,\vee,\wedge\rangle$ 
-  *[[Negative integer lattice]] +  *[[Negative integer chain]] $\langle \mathbb Z^-,\vee,\wedge\rangle$ 
-  *[[Integer lattice]] +  *[[Integer chain]] $\langle \mathbb Z,\vee,\wedge\rangle$ 
-  *[[Rational number chain]]+  *[[Rational number chain]] $\langle \mathbb Q,\vee,\wedge\rangle$