Fuzzy algebraic structure

Remarks

A fuzzy subset $\mu$ is a fuzzy subalgebra if and only if each level set $\mu_t = \set{a \in A : \mu(a) \ge t}$, for $t \in [0, 1]$, is empty or a subalgebra. Indeed, if $a_1, \dots, a_n \in \mu_t$ then $\min_i \mu(a_i) \ge t$; conversely, apply the condition on $\mu_t$ with $t = \min_i \mu(a_i)$. A fuzzy subset that takes only the values $0$ and $1$ is a fuzzy subalgebra exactly when $\set{a : \mu(a) = 1}$ is empty or a subalgebra.

Example: on the group $(\Z, +)$ let $\mu(n) = 1$ for even $n$ and $\mu(n) = 1/2$ for odd $n$. If $m$ or $n$ is odd, the minimum is $1/2$ and $\mu(m + n) \ge 1/2$ holds; if both are even, $m + n$ is even and $\mu(m + n) = 1$. Also $-n$ has the parity of $n$, so $\mu(-n) = \mu(n)$. So $\mu$ is a fuzzy subgroup.

Non-example: on $(\Z, +)$ let $\mu(1) = 1$ and $\mu(n) = 0$ for $n \ne 1$. Then $\mu(1 + 1) = 0 < 1 = \min\set{\mu(1), \mu(1)}$, so $\mu$ is not a fuzzy subgroup.