View source for Semi-lattice

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A commutative idempotent [[semi-group]], that is, a semi-group satisfying the identities $x+y=y+x$ and $x+x=x$. Every semi-lattice $p=\langle p,+\rangle$ can be turned into a [[partially ordered set]] (the partial order $\leq$ is defined by the relation $a\leq b$ if and only if $a+b=b$) in which for any pair of elements there is a least upper bound $\sup\{a,b\}=a+b$. Conversely, every partially ordered set with least upper bounds for every pair of elements is a semi-lattice with respect to the operation $a+b=\sup\{a,b\}$. In this case one says that the partially ordered set is an upper semi-lattice (or a join semi-lattice, or a $\vee$-semi-lattice). A lower semi-lattice, also called a meet semi-lattice or a $\wedge$-semi-lattice, is dually defined as a partially ordered set in which any two elements have a greatest lower bound.



====Comments====
A band is a semi-group every element of which is idempotent (cf. also [[Band of semi-groups]]) (which is a decomposition of a semi-group into sub-semi-groups forming a band). Thus, an upper (lower) semi-lattice defines a commutative band, and conversely.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.H. Clifford,   G.B. Preston,   "The algebraic theory of semigroups" , '''1''' , Amer. Math. Soc.  (1961)  pp. §1.8</TD></TR>
</table>

====Comments====
The free join semilattice on a set $X$ is the set of all finite subsets of $X$ ordered by inclusion.

====References====
<table>
<TR><TD valign="top">[b1]</TD> <TD valign="top">  Peter T. Johnstone ''Sketches of an elephant'' Oxford University Press (2002) ISBN 0198534256 {{ZBL|1071.18001}}</TD></TR>
</table>

[[Category:Order, lattices, ordered algebraic structures]]