The notion of intuitionistic fuzzy positive implicative hyper BCK-ideals of type-1, 2... 8 of hyper BCK-algebras was introduced in 2012 by Durga Prasad, Satyanarayana and Ramesh. In this paper, we investigate some related properties of intuitionistic fuzzy positive implicative hyper BCK-ideals of types-1. We characterize positive implicative Artinian (shortly, PI-Artinian) hyper BCK-algebras of...

First, we construct a bijection between the set of h-vectors and the set of socle-vectors of artinian algebras. As a corollary, we find the minimum codimension that an artinian algebra with a given socle-vector can have. Then, we study the main problem in the paper: determining when there is a unique socle-vector for a given h-vector. We solve the problem completely if the codimension is at mos...

Chain conditions, finiteness conditions, growth conditions, and other forms of finiteness, Noetherian rings and Artinian rings have been systematically studied for commutative rings and algebras since 1959. In pursuit of the deeper results of ideal theory in ordered groupoids (semigroups), it is necessary to study special classes of ordered groupoids (semigroups). Noetherian ordered groupoids (...

The purpose of this note is to characterize the finite Hilbert functions which force all of their artinian algebras to enjoy the Weak Lefschetz Property (WLP). Curiously, they turn out to be exactly those (characterized by Wiebe in [Wi]) whose Gotzmann ideals have the WLP. This implies that, if a Gotzmann ideal has the WLP, then all algebras with the same Hilbert function (and hence lower Betti...

This paper can be seen as a continuation of the works contained in the recent preprints [Za], of the second author, and [Mi], of Juan Migliore. Our results are: 1). There exist codimension three artinian level algebras of type two which do not enjoy the Weak Lefschetz Property (WLP). In fact, for e ≫ 0, we will construct a codimension three, type two h-vector of socle degree e such that all the...

This paper can be seen as a continuation of the works contained in the recent preprints [Za], of the second author, and [Mi], of Juan Migliore. Our results are: 1). There exist codimension three artinian level algebras of type two which do not enjoy the Weak Lefschetz Property (WLP). In fact, for e ≫ 0, we will construct a codimension three, type two h-vector of socle degree e such that all the...