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A commutative idempotent [[Semi-group|semi-group]], that is, a semi-group satisfying the identities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084200/s0842001.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084200/s0842002.png" />. Every semi-lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084200/s0842003.png" /> can be turned into a [[Partially ordered set|partially ordered set]] (the partial order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084200/s0842004.png" /> is defined by the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084200/s0842005.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084200/s0842006.png" />) in which for any pair of elements there is a least upper bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084200/s0842007.png" />. Conversely, every partially ordered set with least upper bounds for every pair of elements is a semi-lattice with respect to the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084200/s0842008.png" />. In this case one says that the partially ordered set is an upper semi-lattice (or a join semi-lattice, or a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084200/s08420010.png" />-semi-lattice). A lower semi-lattice, also called a meet semi-lattice or a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084200/s08420012.png" />-semi-lattice, is dually defined as a partially ordered set in which any two elements have a greatest lower bound.
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{{TEX|done}}{{MSC|06A12}}
  
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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.
  
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====Comments====
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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.
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====References====
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<table>
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<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 {{ZBL|0111.03403}}</TD></TR>
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</table>
  
 
====Comments====
 
====Comments====
A band is a semi-group every element of which is idempotent (cf. also [[Band of semi-groups|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.
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The free join semilattice on a set $X$ is the set of all finite subsets of $X$ ordered by inclusion.
  
 
====References====
 
====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>
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<table>
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<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>
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</table>

Latest revision as of 12:05, 23 November 2023

2020 Mathematics Subject Classification: Primary: 06A12 [MSN][ZBL]

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

[a1] A.H. Clifford, G.B. Preston, "The algebraic theory of semigroups" , 1 , Amer. Math. Soc. (1961) pp. §1.8 Zbl 0111.03403

Comments

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

References

[b1] Peter T. Johnstone Sketches of an elephant Oxford University Press (2002) ISBN 0198534256 Zbl 1071.18001
How to Cite This Entry:
Semi-lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-lattice&oldid=18518
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article