New algebraic structures such as q-rung interval-valued intuitionistic fuzzy set applied to logarithm operator and its extension

We introduce the interval-valued intuitionistic fuzzy set used with the q-rung logarithimic operator (q-LIVIFS). A q-rung interval-valued intuitionistic fuzzy set may be developed by expanding the Pythagorean interval-valued fuzzy set (PIVFS). We discuss the q-logarithimic operator applied to interval-valued intuitionistic fuzzy weighted averaging (q-LIVIFWA), interval-valued intuitionistic fuzzy weighted geometric (q-LIVIFWG), extended q-logarithimic operator applied to interval-valued intuitionistic fuzzy weighted averaging (q-ELIVIFWA), and extended q-logarithimic operator applied to interval-valued intuitionistic fuzzy weighted geometric (q-ELIVIFWG). Several algebraic properties of q-LIVIFSs, such as distributivity, idempotency, and associativity, have been identified.