Irreducible disjoint covering systems (with an application to Boolean algebra) (Q805651)

The authors further develop their lattice geometry approach to the algebra of cosets, namely the part corresponding to irreducible disjoint covering systems of cosets. Here a family \(\{a_ i(n_ i)\); \(1\leq i\leq t\}\) of cosets is called a disjoint covering system (DCS) if it forms a partition of the set of all integers. It is called irreducible if the union of none of its proper subfamilies forms a single coset (possibly 0(1)). The main result of the paper says that if \(\{a_ i(n_ i)\); \(1\leq i\leq t\}\) is an irreducible DCS and \(m=l.c.m.(n_ i;1\leq i\leq t)\) has the prime factors \(p(m)=p_ 1<p_ 2<p_ 3<...<p_{\ell}=P(m)\) then \[ (i)\quad t\geq \max_{1\leq i\leq t}\lambda (n_ i)+(p(m)- 1)P(m);\quad (ii)\quad t\geq \max_{1\leq i\leq t}\lambda (n_ i)+(p_ 1+p_ 2+p_ 3-5), \] where \(\lambda (n)=1+\sum^{k}_{j+1}\alpha_ j(q_ j-1)\) if n has the prime factorization \(n=\prod^{k}_{j=1}q_ j^{\alpha_ j}.\) As a special application to Boolean algebra an improvement of a lower bound for the number of clauses in certain irreducible tautologies is given.