Absorbing element

<h2 id="definition">Definition</h2>
Formally, let (S, •) be a set S with a closed binary operation • on it (known as a <a href="/facts/Magma_(algebra)/9g3vp7Wl">magma</a>). A zero element (or an absorbing/annihilating element) is an element z such that for all s in S, z • s = s • z = z. This notion can be refined to the notions of left zero, where one requires only that z • s = z, and right zero, where s • z = z.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> 
Absorbing elements are particularly interesting for <a href="/facts/Semigroup/rgSdk2kt">semigroups</a>, especially the multiplicative semigroup of a <a href="/facts/Semiring/7cSvWXVQ">semiring</a>. In the case of a semiring with 0, the definition of an absorbing element is sometimes relaxed so that it is not required to absorb 0; otherwise, 0 would be the only absorbing element.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h2 id="properties">Properties</h2>
<ul><li>If a magma has both a left zero z and a right zero z′, then it has a zero, since z = z • z′ = z′.</li>
<li>A magma can have at most one zero element.</li></ul>
<h2 id="examples">Examples</h2>
<ul><li>The most well known example of an absorbing element comes from elementary algebra, where any number multiplied by zero equals zero. Zero is thus an absorbing element.</li>
<li>The zero of any <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> is also an absorbing element. For an element r of a ring R, r0 = r(0 + 0) = r0 + r0, so 0 = r0, as zero is the unique element a for which r − r = a for any r in the ring R. This property holds true also in a <a href="/facts/Rng_(mathematics)/SquNCXKe">rng</a> since multiplicative identity isn't required.</li>
<li><a href="/facts/Floating_point/eIckahxe">Floating point</a> arithmetics as defined in IEEE-754 standard contains a special value called Not-a-Number ("NaN"). It is an absorbing element for every operation; i.e., x + NaN = NaN + x = NaN, x − NaN = NaN − x = NaN, etc.</li>
<li>The set of <a href="/facts/Binary_relations/r9BwGgaK">binary relations</a> over a set X, together with the <a href="/facts/Composition_of_relations/tD8joBlo">composition of relations</a> forms a <a href="/facts/Monoid/6AHve7p8">monoid</a> with zero, where the zero element is the <a href="/facts/Empty_relation/wh0QoWzS">empty relation</a> (<a href="/facts/Empty_set/ffEk3eA6">empty set</a>).</li>
<li>The closed interval H = [0, 1] with x • y = min(x, y) is also a monoid with zero, and the zero element is 0.</li>
<li>More examples:</li></ul>
<table><tbody><tr><th>Domain</th><th colspan="2">Operation</th><th colspan="2">Absorber</th></tr><tr><td><a href="/facts/Real_number/R02gw5Pb">real numbers</a></td><td>⋅</td><td>multiplication</td><td><a href="/facts/0_(number)/4NBRtKHc">0</a></td><td></td></tr><tr><td><a href="/facts/Integer/PUwwrawx">integers</a></td><td></td><td><a href="/facts/Greatest_common_divisor/kNFvBuKl">greatest common divisor</a></td><td>1</td><td></td></tr><tr><td>n-by-n square <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a></td><td></td><td><a href="/facts/Matrix_multiplication/lYDB6Ro2">matrix multiplication</a></td><td></td><td><a href="/facts/Zero_matrix/uqbGFFww">matrix of all zeroes</a></td></tr><tr><td rowspan="2"><a href="/facts/Extended_real_number_line/65Gs7QS0">extended real numbers</a></td><td></td><td>minimum/infimum</td><td>−∞</td><td></td></tr><tr><td></td><td>maximum/supremum</td><td>+∞</td><td></td></tr><tr><td><a href="/facts/Set_(mathematics)/BfucfHMq">sets</a></td><td>∩</td><td>intersection</td><td>∅</td><td><a href="/facts/Empty_set/ffEk3eA6">empty set</a></td></tr><tr><td>subsets of a set M</td><td>∪</td><td>union</td><td>M</td><td></td></tr><tr><td rowspan="2"><a href="/facts/Boolean_algebra/VkCToAmg">Boolean logic</a></td><td>∧</td><td><a href="/facts/Logical_conjunction/62uyyn1d">logical and</a></td><td>⊥</td><td>falsity</td></tr><tr><td>∨</td><td><a href="/facts/Logical_disjunction/jbI2EGlM">logical or</a></td><td>⊤</td><td>truth</td></tr></tbody></table>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Absorbing_set/GCtjsUuX">Absorbing set</a> – Set that can be "inflated" to reach any point</li>
<li><a href="/facts/Annihilator_(disambiguation)/TvNld8FH">Annihilator (disambiguation)</a></li>
<li><a href="/facts/Annihilator_(ring_theory)/aTaHR5W7">Annihilator (ring theory)</a> – Ideal that maps to zero a subset of a module</li>
<li><a href="/facts/Idempotent_(ring_theory)/RE6AkdQd">Idempotent (ring theory)</a> – In mathematics, element that equals its square – an element x of a ring such that x2 = x</li>
<li><a href="/facts/Identity_element/9nss3xHY">Identity element</a> – Specific element of an algebraic structure</li>
<li><a href="/facts/Null_semigroup/eyRwHwrM">Null semigroup</a> – semigroup with an absorbing element, called zero, in which the product of any two elements is zeroPages displaying wikidata descriptions as a fallback</li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Howie, John M. (1995). Fundamentals of Semigroup Theory. <a href="/facts/Clarendon_Press/CYyTzzWG">Clarendon Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-851194-9.</li>
<li>Kilp, M.; Knauer, U.; Mikhalev, A.V. (2000), "Monoids, Acts and Categories with Applications to Wreath Products and Graphs", De Gruyter Expositions in Mathematics, 29, Walter de Gruyter, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-11-015248-7</li>
<li>Golan, Jonathan S. (1999). Semirings and Their Applications. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7923-5786-8.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://planetmath.org/absorbingelement">Absorbing element</a> at PlanetMath</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Howie 1995, pp. 2–3 - Howie, John M. (1995). Fundamentals of Semigroup Theory. Clarendon Press. ISBN 0-19-851194-9. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Kilp, Knauer & Mikhalev 2000, pp. 14–15 - Kilp, M.; Knauer, U.; Mikhalev, A.V. (2000), "Monoids, Acts and Categories with Applications to Wreath Products and Graphs", De Gruyter Expositions in Mathematics, 29, Walter de Gruyter, ISBN 3-11-015248-7 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Kilp, Knauer & Mikhalev 2000, pp. 14–15 - Kilp, M.; Knauer, U.; Mikhalev, A.V. (2000), "Monoids, Acts and Categories with Applications to Wreath Products and Graphs", De Gruyter Expositions in Mathematics, 29, Walter de Gruyter, ISBN 3-11-015248-7 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Golan 1999, p. 67 - Golan, Jonathan S. (1999). Semirings and Their Applications. Springer. ISBN 0-7923-5786-8. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
</ol>

Absorbing element open-in-new

Absorbing element