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+ | =====Rectangular bands===== | ||
+ | Abbreviation: **RBand** | ||
+ | ====Definition==== | ||
+ | A \emph{rectangular band} is a [[bands]] $\mathbf{B}=\langle B,\cdot | ||
+ | \rangle $ such that | ||
+ | |||
+ | $\cdot $ is rectangular: $x\cdot y\cdot x=x$. | ||
+ | ====Definition==== | ||
+ | A \emph{rectangular band} is a [[bands]] $\mathbf{B}=\langle B,\cdot | ||
+ | \rangle $ such that | ||
+ | |||
+ | $x\cdot y\cdot z=x\cdot z$. | ||
+ | ==Morphisms== | ||
+ | Let $\mathbf{B}$ and $\mathbf{C}$ be rectangular 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==== | ||
+ | [[Left-zero semigroups]] | ||
+ | |||
+ | [[Right-zero semigroups]] | ||
+ | |||
+ | ====Superclasses==== | ||
+ | [[Normal bands]] | ||
+ | |||
+ | |||
+ | ====References==== | ||
+ | |||
+ | [(Ln19xx> | ||
+ | )] |
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