Greatest element and least element

Greatest element and least element

In <a href="/en/Mathematics" title="Mathematics" class="wl">mathematics</a>, especially in <a href="/en/Order_theory" title="Order theory" class="wl">order theory</a>, the greatest element of a subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">
 <semantics>
 <mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>S</mi>
 </mstyle>
 </mrow>
 <annotation encoding="application/x-tex">{\displaystyle S}</annotation>
 </semantics>
</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="S"> of a <a href="/en/Partially_ordered_set" title="Partially ordered set" class="wl">partially ordered set</a> (poset) is an element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">
 <semantics>
 <mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>S</mi>
 </mstyle>
 </mrow>
 <annotation encoding="application/x-tex">{\displaystyle S}</annotation>
 </semantics>
</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="S"> that is greater than every other element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">
 <semantics>
 <mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>S</mi>
 </mstyle>
 </mrow>
 <annotation encoding="application/x-tex">{\displaystyle S}</annotation>
 </semantics>
</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="S">. The term least element is defined <a href="/en/Duality_(order_theory)" title="Duality (order theory)" class="wl">dually</a>, that is, it is an element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">
 <semantics>
 <mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>S</mi>
 </mstyle>
 </mrow>
 <annotation encoding="application/x-tex">{\displaystyle S}</annotation>
 </semantics>
</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="S"> that is smaller than every other element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S.}">
 <semantics>
 <mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>S</mi>
 <mo>.</mo>
 </mstyle>
 </mrow>
 <annotation encoding="application/x-tex">{\displaystyle S.}</annotation>
 </semantics>
</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23bbb1f0f6ebdfa78b4fed06049640f7386bb44b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.146ex; height:2.176ex;" alt="S.">


<div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none" about="#mwt3">Element ≥ (or ≤) each other element</div>
<figure class="mw-default-size wiki-thumb"><a href="/en/File:Lattice_of_the_divisibility_of_60_narrow_1,2,3,4.svg" class="mw-file-description image media-thumb" data-hash="File:Lattice_of_the_divisibility_of_60_narrow_1,2,3,4.svg"><img loading="lazy" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Lattice_of_the_divisibility_of_60_narrow_1%2C2%2C3%2C4.svg/220px-Lattice_of_the_divisibility_of_60_narrow_1%2C2%2C3%2C4.svg.png" decoding="async" data-file-width="183" data-file-height="222" data-file-type="drawing" height="267" width="220" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Lattice_of_the_divisibility_of_60_narrow_1%2C2%2C3%2C4.svg/330px-Lattice_of_the_divisibility_of_60_narrow_1%2C2%2C3%2C4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Lattice_of_the_divisibility_of_60_narrow_1%2C2%2C3%2C4.svg/440px-Lattice_of_the_divisibility_of_60_narrow_1%2C2%2C3%2C4.svg.png 2x" alt="Lattice_of_the_divisibility_of_60_narrow_1%2C2%2C3%2C4.svg"></a><figcaption><a href="/en/Hasse_diagram" title="Hasse diagram" class="wl">Hasse diagram</a> of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
 <semantics>
 <mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>P</mi>
 </mstyle>
 </mrow>
 <annotation encoding="application/x-tex">{\displaystyle P}</annotation>
 </semantics>
</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="P"> of <a href="/en/Divisor" title="Divisor" class="wl">divisors</a> of 60, partially ordered by the relation "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
 <semantics>
 <mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>x</mi>
 </mstyle>
 </mrow>
 <annotation encoding="application/x-tex">{\displaystyle x}</annotation>
 </semantics>
</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"> divides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">
 <semantics>
 <mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>y</mi>
 </mstyle>
 </mrow>
 <annotation encoding="application/x-tex">{\displaystyle y}</annotation>
 </semantics>
</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="y">". The red subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\{1,2,3,4\}}">
 <semantics>
 <mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>S</mi>
 <mo>=</mo>
 <mo fence="false" stretchy="false">{</mo>
 <mn>1</mn>
 <mo>,</mo>
 <mn>2</mn>
 <mo>,</mo>
 <mn>3</mn>
 <mo>,</mo>
 <mn>4</mn>
 <mo fence="false" stretchy="false">}</mo>
 </mstyle>
 </mrow>
 <annotation encoding="application/x-tex">{\displaystyle S=\{1,2,3,4\}}</annotation>
 </semantics>
</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/651b92f894e4495248a2189441f5cfb989f55848" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:14.674ex; height:2.843ex;" alt="{\displaystyle S=\{1,2,3,4\}}"> has two maximal elements, viz. 3 and 4, and one minimal element, viz. 1, which is also its least element.</figcaption></figure>

In mathematics, especially in order theory, the greatest element of a subset 
  
    
      
        S
      
    
    {\displaystyle S}
  
 of a partially ordered set is an element of 
  
    
      
        S
      
    
    {\displaystyle S}
  
 that is greater than every other element of 
  
    
      
        S
      
    
    {\displaystyle S}
  
. The term least element is defined dually, that is, it is an element of 
  
    
      
        S
      
    
    {\displaystyle S}
  
 that is smaller than every other element of 
  
    
      
        S
        .
      
    
    {\displaystyle S.}
  
 