# Differences

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+ | =====Normal bands===== | ||

+ | Abbreviation: **NBand** | ||

+ | ====Definition==== | ||

+ | A \emph{normal band} is a [[bands]] $\mathbf{B}=\langle B,\cdot | ||

+ | \rangle $ such that | ||

+ | |||

+ | $\cdot $ is normal: $x\cdot y\cdot z\cdot x=x\cdot z\cdot y\cdot x$. | ||

+ | ==Morphisms== | ||

+ | Let $\mathbf{B}$ and $\mathbf{C}$ be normal bands. A morphism from $\mathbf{B}$ | ||

+ | to $\mathbf{C}$ is a function $h:B\rightarrow C$ that is a homomorphism: | ||

+ | |||

+ | $h(xy)=h(x)h(y)$ | ||

+ | |||

+ | ====Examples==== | ||

+ | |||

+ | |||

+ | ====Basic results==== | ||

+ | |||

+ | ====Properties==== | ||

+ | ^[[Classtype]] |variety | | ||

+ | ^[[Equational theory]] |decidable in polynomial time | | ||

+ | ^[[Quasiequational theory]] | | | ||

+ | ^[[First-order theory]] | | | ||

+ | ^[[Locally finite]] |yes | | ||

+ | ^[[Residual size]] | | | ||

+ | ^[[Congruence distributive]] | | | ||

+ | ^[[Congruence modular]] | | | ||

+ | ^[[Congruence n-permutable]] | | | ||

+ | ^[[Congruence regular]] | | | ||

+ | ^[[Congruence uniform]] | | | ||

+ | ^[[Congruence extension property]] | | | ||

+ | ^[[Definable principal congruences]] | | | ||

+ | ^[[Equationally def. pr. cong.]] | | | ||

+ | ^[[Amalgamation property]] | | | ||

+ | ^[[Strong amalgamation property]] | | | ||

+ | ^[[Epimorphisms are surjective]] | | | ||

+ | ====Finite members==== | ||

+ | |||

+ | $\begin{array}{lr} | ||

+ | f(1)= &1\\ | ||

+ | f(2)= &\\ | ||

+ | f(3)= &\\ | ||

+ | f(4)= &\\ | ||

+ | f(5)= &\\ | ||

+ | f(6)= &\\ | ||

+ | f(7)= &\\ | ||

+ | \end{array}$ | ||

+ | |||

+ | ====Subclasses==== | ||

+ | [[Rectangular bands]] | ||

+ | |||

+ | ====Superclasses==== | ||

+ | [[Bands]] | ||

+ | |||

+ | |||

+ | ====References==== | ||

+ | |||

+ | [(Ln19xx> | ||

+ | )] |

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