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True if and only if all subgroups are trivial. That is, no two distinct elements are both \(\mathcal{L}\)-related and \(\mathcal{R}\)-related.

True if and only if all elements are idempotent.

True if and only if \(xy=yx\) for all \(x\) and \(y\).

True if and only if all regular \(\mathcal{D}\)-classes are aperiodic semigroups. Equivalently: all regular elements are idempotent.

True if and only if no two distinct elements are \(\mathcal{J}\)-related. Equivalently: both \(\mathcal{L}\)-trivial and \(\mathcal{R}\)-trivial.

True if and only if no two distinct elements are \(\mathcal{L}\)-related.

True if and only if the only idempotent element is zero.

True if and only if no two distinct elements are \(\mathcal{L}\)-related.

True if and only if there is but a single element.

True if and only if each of the localSubmonoids satisfies the given proposition.