The factorization and representation of lattices
George Markowsky · Transactions of the American Mathematical Society · 1975
For a complete lattice L L , in which every element is a join of completely join-irreducibles and a meet of completely meet-irreducibles (we say L L is a jm-lattice ) we define the poset of irreducibles P ( L ) P(L) to be the poset (of height one) J ( L ) ∪ M ( L ) ( J ( L ) J(L) \cup M(L)(J(L) is the set of completely join-irreducibles and M ( L ) M(L) is the set of completely meet-irreducibles) ordered as follows: a > P ( L ) b a{ > _{P(L)}}b if and only if a ∈ J ( L ) , b ∈ M ( L ) a \in J(L),b \in M(L) , and a ⪇ L b a leqslant { _L}b . For a jm-lattice L L , the automorphism groups of L L and P ( L )