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semilattices [2010/07/29 18:30]
127.0.0.1 external edit
semilattices [2020/03/24 17:24] (current)
pnotthesamejipsen
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This definition shows that semilattices form a variety. This definition shows that semilattices form a variety.
 +==Morphisms==
 +Let $\mathbf{S}$ and $\mathbf{T}$ be semilattices. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\to T$ that is a homomorphism:
 +
 +$h(xy)=h(x)h(y)$
 +
 +====Definition====
 +A \emph{join-semilattice} is a structure $\mathbf{S}=\langle S,\leq,\vee\rangle$, where $\vee $ is an infix binary operation, called the \emph{join}, such that
 +
 +$\leq $ is a partial order,
 +
 +$x\leq y\implies x\vee z\leq y\vee z$ and $z\vee x\leq z\vee y$,
 +
 +$x\le x\vee y$ and $y\leq x\vee y$,
 +
 +$x\vee x\leq x$.
 +This definition shows that semilattices form a partially-ordered variety.
====Definition==== ====Definition====
A \emph{join-semilattice} is a structure $\mathbf{S}=\langle S,\vee A \emph{join-semilattice} is a structure $\mathbf{S}=\langle S,\vee
-\rangle $, where $\vee $ is an infix binary operation, called the $\emph{join}$, such that+\rangle $, where $\vee $ is an infix binary operation, called the \emph{join}, such that
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====Definition==== ====Definition====
A \emph{meet-semilattice} is a structure $\mathbf{S}=\langle S,\wedge A \emph{meet-semilattice} is a structure $\mathbf{S}=\langle S,\wedge
-\rangle $, where $\wedge $ is an infix binary operation, called the $\emph{meet}$, such that+\rangle $, where $\wedge $ is an infix binary operation, called the \emph{meet}, such that
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$x\wedge y$ is the greatest lower bound of $\{x,y\}$. $x\wedge y$ is the greatest lower bound of $\{x,y\}$.
-==Morphisms== 
-Let $\mathbf{S}$ and $\mathbf{T}$ be semilattices. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:Sarrow T$ that is a homomorphism:  
- 
-$h(xy)=h(x)h(y)$ 
====Examples==== ====Examples====
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^[[Congruence uniform]]  |no | ^[[Congruence uniform]]  |no |
^[[Congruence extension property]]  | | ^[[Congruence extension property]]  | |
-^[[Definable principal congruences]]  | | +^[[Definable principal congruences]]  |yes
-^[[Equationally def. pr. cong.]]  | |+^[[Equationally def. pr. cong.]]  |yes |
^[[Amalgamation property]]  |yes | ^[[Amalgamation property]]  |yes |
^[[Strong amalgamation property]]  |yes | ^[[Strong amalgamation property]]  |yes |
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Algebra Universalis, Algebra Universalis,
\textbf{48}2002,43--53[[MRreview]] \textbf{48}2002,43--53[[MRreview]]
- 
-[[Search for finite semilattices]] 
====Subclasses==== ====Subclasses====
-[[One-element algebras]]  
- 
[[Semilattices with zero]] [[Semilattices with zero]]
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====Superclasses==== ====Superclasses====
 +[[Bands]]
 +
[[Commutative semigroups]] [[Commutative semigroups]]