Summary. We present a Mizar formalization of chapter 4.4 of [8] devoted to special orderings in additive monoids to be used for ordering terms in multivariate polynomials. We have extended the treatment to the case of infinite number of variables. It turns out that in such case admissible orderings are not necessarily well orderings. MML Identifier:BAGORDER. WWW:http://mizar.org/JFM/Vol14/bagorder.html

Summary. In this article we define noetherian and co-noetherian lattices and show how some properties concerning upper and lower neighbours, irreducibility and density can be improved when restricted to these kinds of lattices. In addition we define atomic lattices.

this paper. 1. Preliminaries Let X be a set. Let us observe that X has non empty elements if and only if: (Def. 1) 0 / # X. We introduce X is without zero as a synonym of X has non empty elements. We introduce X has zero as an antonym of X has non empty elements. Let us note that R has zero and N has zero. Let us note that there exists a set which is non empty and without zero and there exists a set which is non empty and has zero. Let us observe that there exists a subset of R which is non empty and without zero and there exists a subset of R which is non empty and has zero. The following proposition is true (1) For every set F such that F is non empty and #-linear and has non empty elements holds F is centered. Let