algebras

Algebras

2-element boolean algebra $\mathbb B_2=\langle\{0,1\},\vee,0,\wedge,1,'\rangle$
2-element chain $\mathbb C_2=\langle\{0,1\},\vee,\wedge\rangle$
2-element cyclic group $\mathbb Z_2=\langle\{0,1\},+_2,-_2,0\rangle$
2-element field $\mathbb F_2=\langle\{0,1\},+_2,-_2,0,\cdot_2,1\rangle$
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 $\mathbb S_2=\langle\{0,1\},\cdot\rangle$
2-element set $\mathbf 2=\langle\{0,1\}\rangle$

3-element chain $\mathbb C_3=\langle\{0,1,2\},\vee,\wedge\rangle$
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 semilattices $\mathbb S_{3,0}=\langle\{0,1,2\},\cdot\rangle$, $\mathbb S_{3,1}=\langle\{0,1,2\},\min\rangle$

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 cyclic group $\mathbb Z_4=\langle\{0,1,2,3\},+_4,-_4,0\rangle$
4-element distributive lattice $(\mathbb C_2)^2=\langle\{(0,0),(0,1),(1,0),(1,1)\},\vee,\wedge\rangle$
4-element field $\mathbb F_4=\langle\{0,1,x,x+1\},+_2,-_2,0,\cdot,1\rangle\cong\mathbb F_2[x]/\langle x^2+x+1\rangle$
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 $\mathbb Z_{4,1}=\langle\{0,1,2,3\},+_4,-_4,0,\cdot_4,1\rangle$, $(\mathbb F_2)^2=\langle\{(0,0),(0,1),(1,0),(1,1)\},+_2,-_2,(0,0),\cdot,(1,1)\rangle$

6-element nonabelian group $S_3=\langle\{(),(12),(13),(23),(123),(132)\},\circ,{}^{-1},()\rangle$

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 cyclic group $\mathbb Z_n=\langle\{0,1,2,\ldots,n-1\},+_n,-_n,0\rangle$
p^k-element field $\mathbb F_{p^k}=\langle\{0,1,\ldots,p-1,x,\ldots\},+_p,-_p,0,\cdot,1\rangle\cong\mathbb F_p[x]/\langle f(x)\rangle$