normal_bands - MathStructures

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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>
)]

Trace: • idempotent_semirings_with_zero • bands • flw-algebras • finite_commutative_binars • solvable_groups • distributive_lattices_with_operators • modal_algebras • m-sets • pocrims • m-zeroid