| Summary: | In this paper, first, we state an operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mi>R</mi></msub></semantics></math></inline-formula> on an ordered semihyperring <i>R</i>. We show that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi><mo>:</mo><mi>R</mi><mo>⟶</mo><mi>T</mi></mrow></semantics></math></inline-formula> is a monomorphism and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>⊆</mo><mi>R</mi></mrow></semantics></math></inline-formula>, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mi>T</mi></msub><mrow><mo>(</mo><mi>φ</mi><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mi>φ</mi><mrow><mo>(</mo><msub><mi>L</mi><mi>R</mi></msub><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Afterward, hyperatom elements in ordered semihyperrings are defined and some results in this respect are investigated. Denote by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> the set of all hyperatoms of <i>R</i>. We prove that if <i>R</i> is a finite ordered semihyperring and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>R</mi><mo>|</mo><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>, then for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mi>R</mi><mo>\</mo><mo>{</mo><mn>0</mn><mo>}</mo></mrow></semantics></math></inline-formula>, there exists <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>h</mi><mi>q</mi></msub><mo>∈</mo><msup><mi>A</mi><mo>*</mo></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>=</mo><mi>A</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>\</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>h</mi><mi>q</mi></msub><mo>≤</mo><mi>q</mi></mrow></semantics></math></inline-formula>. Finally, we study the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mi>R</mi></msub></semantics></math></inline-formula>-graph of an ordered semihyperring and give some examples. Furthermore, we show that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi><mo>:</mo><mi>R</mi><mo>⟶</mo><mi>T</mi></mrow></semantics></math></inline-formula> is an isomorphism, <i>G</i> is the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mi>R</mi></msub></semantics></math></inline-formula>-graph of <i>R</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>G</mi><mo>′</mo></msup></semantics></math></inline-formula> is the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mi>T</mi></msub></semantics></math></inline-formula>-graph of <i>T</i>, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>≅</mo><msup><mi>G</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula>.
|
|---|