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Let $D$ be an integral domain with quotient field $K$. The $b$-operation thatassociates to each nonzero $D$-submodule $E$ of $K$, $E^b := \bigcap\{EV \mid Vvaluation overring of D\}$, is a semistar operation that plays an importantrole in many questions of ring theory (e.g., if $I$ is a nonzero ideal in $D$,$I^b$ coincides with its integral closure). In a first part of the paper, westudy the integral domains that are $b$-Noetherian (i.e., such that, for eachnonzero ideal $I$ of $D$, $I^b = J^b$ for some a finitely generated ideal $J$of $D$). For instance, we prove that a $b$-Noetherian domain has Noetherianspectrum and, if it is integrally closed, is a Mori domain, but integrallyclosed Mori domains with Noetherian spectra are not necessarily $b$-Noetherian.We also characterize several distinguished classes of $b$-Noetherian domains.In a second part of the paper, we study more generally the e.a.b. semistaroperation of finite type $\star_a$ canonically associated to a given semistaroperation $\star$ (for instance, the $b$-operation is the e.a.b. semistaroperation of finite type canonically associated to the identity operation).These operations, introduced and studied by Krull, Jaffard, Gilmer andHalter-Koch, play a very important role in the recent generalizations of theKronecker function ring. In particular, in the present paper, we classifyseveral classes of integral domains having some of the fundamental operations$d$, $t$, $w$ and $v$ equal to some of the canonically associated e.a.b.operations $b$, $t_a$, $w_a$ and $v_a$.

On some classes of integral domains defined by Krull's $\boldsymbol{a.b.}$ operations