Given multiplicatively-regular elements <mml:math alttext="$a$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>a</mml:mi></mml:math> and <mml:math alttext="$b$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>b</mml:mi></mml:math> in a semiring
<mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math>, and given an element <mml:math alttext="$c$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>c</mml:mi></mml:math> of <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math>, we find a complete set of solutions to the equation <mml:math alttext="$aXb=c$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>a</mml:mi><mml:mi>X</mml:mi><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:math>. This result is then extended to equations over matrix semirings.

One-sided complements and solutions of the equation <mml:math alttext="$aXb=c$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>a</mml:mi><mml:mi>X</mml:mi><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:math> in semirings

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International Journal of Mathematics and Mathematical Sciences

One-sided complements and solutions of the equation aXb=c in semirings

Abstract

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