On some non-rational congruences on free commutative semigroups (Q688963)

Let \(F_ n\) denote the free commutative semigroup over the \(n\) generators \(a_ 1,a_ 2,\dots,a_ n\). A congruence \(\rho\) on \(F_ n\) is called an IF-archimedean congruence on \(F_ n\) if \(F_ n/\rho\) is a commutative archimedean idempotent-free semigroup. If \(F_ n/\rho\) is in addition cancellative, \(\rho\) is called a CIF-archimedean congruence. In this paper the author studies the lattices of IF-archimedean congruences under a certain restrictive condition. He introduces \(p\)-linear congruences and primary congruences as follows. Let \(N\) be the set of positive integers and let \(\lambda = (l_ 1,\dots,l_ n) \in N^ n\). Define a congruence \(\rho_ \lambda\) on \(F_ n\) by \((\sum^ n_{i = 1} x_ i a_ i) \rho_ \lambda(\sum^ n_{i = 1} y_ ia_ i)\) if \(\sum^ n_{i = 1} l_ i x_ i = \sum^ n_{i = 1} l_ i y_ i\). We call a congruence \(\tau\) on \(F_ n\) a \(p\)-linear congruence if \(\tau = \rho_ \lambda\) for some \(\lambda\). Let \(m_ i\), \(p_ j\) be positive integers, \(1 \leq i \leq n-1\), \(2 \leq j \leq n\). A primitive congruence is defined to be the congruence on \(F_ n\) defined by \(m_ i a_ i = p_{i+1} a_{i+1}\), \(1 \leq i \leq n-1\). The following is an outline of the results. The set \(L_ \lambda\) of IF- archimedean congruences on \(F_ n\) contained in \(\rho_ \lambda\) forms a sublattice of all congruences contained in \(\rho_ \lambda\). Given positive integers \(m_ i\), \(p_ j\), \(1 \leq i \leq n-1\), \(2 \leq j \leq n\). Let \(\rho\) be the primitive congruence defined by \(m_ i a_ i = p_{i+1} a_{i+1}\). Then there exists a cancellative semigroup \(T(n,m_ i,p_{i+1})\) such that \(F_ n/\rho \cong T(n,m_ i,p_{i+1})\), hence \(\rho\) is an IF-archimedean congruence. Let \(P_ \lambda\) be the set of primitive congruences contained in \(L_ \lambda\), then \((P_ \lambda,\vee,\wedge)\) is a lattice. (If \(\rho\) and \(\tau\) are \(\lambda\)-congruences associated with \((m_ 1 ,\dots, m_{n-1})\) and \((m_ 1',\dots,m_{n-1}')\), then \(\rho \wedge \tau\) is associated with \((m_ 1^{\prime\prime},\dots, m_{n-1}^{\prime\prime})\), \(m_ i^{\prime\prime} = \text{lcm}(m_ i,m_ i')\), \(1 \leq i \leq n-1\) and \(\rho \vee \tau\) is associated with \((d_ 1, \dots d_{n-1})\), \(d_ i = \text{gcd}(m_ i,m_ i')\), \(1 \leq i \leq n-1\).) The author also obtains a necessary and sufficient condition for a primitive congruence to equal a \(p\)-linear congruence. Finally he treats the case \(n = 2\). It is shown that every CIF-archimedean congruence on \(F_ 2\) is primitive, and he shows how to obtain all CIF-archimedean congruences on \(F_ 2\).