bands - MathStructures

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=====Bands=====

====Definition====
A \emph{band} is a [[semigroup]] $\mathbf{B}=\langle B,\cdot
\rangle $ such that

$\cdot $ is idempotent:  $x\cdot x=x$.

==Morphisms==
Let $\mathbf{B}$ and $\mathbf{C}$ be bands. A morphism from $\mathbf{B}$
to $\mathbf{C}$ is a function $h:B\to C$ 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]]  |no |
^[[Congruence modular]]  |no |
^[[Congruence n-permutable]]  |no |
^[[Congruence regular]]  |no |
^[[Congruence uniform]]  |no |
^[[Congruence extension property]]  |no |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  |no |
^[[Strong amalgamation property]]  |no |
^[[Epimorphisms are surjective]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &3\\
f(3)= &10\\
f(4)= &46\\
f(5)= &251\\
f(6)= &1682\\
f(7)= &13213\\
\end{array}$

see also [[finite bands]] and http://www.research.att.com/projects/OEIS?Anum=A058112


====Subclasses====
[[Rectangular bands]] 

[[Semilattices]] 

====Superclasses====
[[Semigroups]] 


====References====

[(Ln19xx>
)]

Trace: • commutative_residuated_lattice-ordered_semigroups • implicative_lattices • frames • commutative_ordered_monoids • normed_vector_spaces • commutative_partially_ordered_monoids • monounary_algebras • unique_factorization_domains • mv-algebras • small_categories