We study ergodic Sym(N)-invariant probability measures on the space of L-structures with domain N. We call such measures “ergodic structures”. In particular, we are interested in the properly ergodic case, in which no isomorphism class has measure 1. A Morley–Scott analysis shows that proper ergodicity can always be explained by a splitting of measure over continuum-many types in a countable fr...

We introduce a proper display calculus for (non-distributive) Lattice Logic which is sound, complete, conservative, and enjoys cut-elimination and subformula property. Properness (i.e. closure under uniform substitution of all parametric parts in rules) is the main interest and added value of the present proposal, and allows for the smoothest Belnap-style proof of cut-elimination. Our proposal ...

We introduce proper display calculi for intuitionistic, bi-intuitionistic and classical linear logics with exponentials, which are sound, complete, conservative, and enjoy cut-elimination and subformula property. Based on the same design, we introduce a variant of Lambek calculus with exponentials, aimed at capturing the controlled application of exchange and associativity. Properness (i.e. clo...

We introduce the class of “right almost V-rings” which is properly between the classes of right V-rings and right good rings. A ring R is called a right almost V-ring if every simple R-module is almost injective. It is proved that R is a right almost V-ring if and only if for every R-module M, any complement of every simple submodule of M is a direct summand. Moreover, R is a right almost V-rin...

  Small groups that gather to learn a specific topic or solve a problem bring golden teaching opportunities with extraordinary outcomes if being properly used. This happens since teaching in small groups is among the most effective approaches in teaching that can be highly efficient in learning. Small groups that may form at the very least with two members and can expand up to thirty members h...