Monoids for which condition \((P)\) acts are projective
Let \(S\) denote a monoid. The author defines a sequence \(q_0,q_1,\ldots\) of elements of \(S\) to be left collapsing if \(q_i=q_iq_{i-1}\) for all \(i\geq 1\) and he says that a submonoid \(M\) of \(S\) is strongly left collapsible if it has a generating set \(F\) of idempotents with the property that for all \(x,y\) in \(M\) there is an element \(f\) of \(F\) such that \(fx=fy\). A monoid is left collapsible if for any \(p,q\in S\) there exists \(r\in S\) such that \(rp=rq\). The following conditions are used: \(FP_1\): for every left collapsing sequence \(q_0,q_1,\ldots\) of elements of \(S\) there exists \(m\) with \(q_m=q_mq_i\) for all \(i\geq 0\). \(FP_2\): every strongly left collapsible submonoid of \(S\) contains a left zero. \(A\): is the ascending chain condition for cyclic subacts. A monoid \(S\) is said to be right reversible if any two principal left ideals of \(S\) intersect. The author proves (partially citing): 1. The following are equivalent: (i) All cyclic right \(S\)-acts that satisfy condition \((P)\) are projective, (ii) every right reversible submonoid of \(S\) contains a left zero, (iii) \(S\) is aperiodic and satisfies \(FP_1\) and \(FP_2\), (iv) \(S\) is aperiodic and satisfies \(FP_2\), (v) \(S\) is aperiodic and every left collapsible submonoid of \(S\) contains a left zero. 2. The following are equivalent. (i) All cyclic right \(S\)-acts that satisfy condition \((P)\) are strongly flat. (ii) \(S\) is aperiodic. 3. All right \(S\)-acts that satisfy condition \((P)\) are projective if and only if \(S\) satisfies condition \((A)\) and every right reversible submonoid of \(S\) contains a left zero. 4. If \(S\) satisfies condition \((A)\) and is aperiodic, then all right \(S\)-acts that satisfy condition \((P)\) are strongly flat. An \(S\)-act \(A\) is said to be locally cyclic if every finitely generated subact is contained within a cyclic subact. Clearly, this is equivalent to saying that for all \(x,y\in A\), there exists \(z\) in \(A\) such that \(x,y\in zS\). 5. A right \(S\)-act X that satisfies condition \((P)\) is indecomposable if and only if it is locally cyclic. 6. Let \(X\) be a locally cyclic right \(S\)-act. Then \(X\) is strongly flat if and only if \(X\) satisfies condition \((E)\). 7. Every right \(S\)-act that satisfies condition \((E)\) is strongly flat if and only if the indecomposable right \(S\)-acts that satisfy condition \((E)\) are locally cyclic.
scientific article; zbMATH DE number 1491100