# Differences

This shows you the differences between two versions of the page.

semilattices [2010/07/29 18:30]
127.0.0.1 external edit
semilattices [2020/03/24 17:24] (current)
pnotthesamejipsen
Line 20: Line 20:
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
Line 32: Line 48:
====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
Line 39: Line 55:
$x\wedge y$ 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====
Line 64: Line 76:
^[[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 |
Line 97: Line 109:
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]]
Line 108: Line 116:
====Superclasses==== ====Superclasses====
+[[Bands]]
+
[[Commutative semigroups]] [[Commutative semigroups]]

##### Toolbox 