  Rosenfeld defined a fuzzy subgroup of group $G$ as a fuzzy subset of $G$ with
two special conditions attached\cite{Rosenfeld1971Fuzzysubgroups}. In this
paper, we introduce the fuzzy $t$-norms and vague $t$-norms. The unit interval
with a $t$-norm or a $t$-conorm is a special monoid, so we mainly talk about
fuzzy subsets of monoids and vague monoids. Firstly, we generalize some
properties of $t$-norm to fuzzy $t$-subnorm, so that we can analyze and
classify the fuzzy $t$-norms. Further, we explore specific research on these
properties of the vague $t$-norm. In addition, the concept of lattice is
introduced, and then the present conclusions are extended to bounded lattices.
Finally, we define the concept of fuzzy monoids by aggregate functions,
uninorms, nullnorms and draw the relevant results.
