| Summary: | The aim of this paper is to determine several saturated classes of structurally regular semigroups. First, we show that structurally <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></semantics></math></inline-formula>-regular semigroups are saturated in a subclass of semigroups for any pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></semantics></math></inline-formula> of positive integers. We also demonstrate that, for all positive integers <i>n</i> and <i>k</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></semantics></math></inline-formula>, the variety of structurally <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>-left seminormal bands is saturated in the variety of structurally <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>-bands. As a result, in the category of structurally <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>-bands, epis from structurally <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>-left seminormal bands is onto.
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