# Differences

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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]] | ||

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