Łukasiewicz fuzzy sub-hoops

With the purpose of applying the Łukasiewicz fuzzy set to sub-hoop in hoops, the concept of Łukasiewicz fuzzy sub-hoop is introduced, and its properties are investigated. The relationship between fuzzy sub-hoop and Łukasiewicz fuzzy sub-hoop are discussed. Conditions for Łukasiewicz fuzzy set to be a Łukasiewicz fuzzy sub-hoop are provided, and characterizations of Łukasiewicz fuzzy sub-hoop are displayed. Conditions under which <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow> <mml:mo class="MathClass-rel" stretchy="false">∈</mml:mo></mml:mrow></mml:math>-set, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>q</mml:mi></mml:math>-set, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>O</mml:mi></mml:math>-set can be sub-hoops are investigated.