By our count, 245 projective planes of order less than 32 are currently known. This list is dominated by the 193 known planes of order 25. Most of these are either translation planes or Hughes planes, or planes obtained from these by the well-known process of repeatedly dualizing and deriving. We describe two new planes obtainable by the quite different method of ‘lifting quotients’. 2000 Mathe...

The notion of toric face rings generalizes both Stanley-Reisner rings and affine semigroup rings, and has been studied by Bruns, Römer, et.al. Here, we will show that, for a toric face ring R, the “graded” Matlis dual of a Cěch complex gives a dualizing complex. In the most general setting, R is not a graded ring in the usual sense. Hence technical argument is required.

Hierarchical distance-based fuzzy multi-criteria group decision making was served as a tool to evaluate the drinking water supply systems of Qom, a semi-arid city located in central part of Iran. A list of aspects consisting of 6 criteria and 35 sub-criteria were evaluated based on a linguistic term set by five decision-makers. Four water supply alternatives including "Public desalinated distri...

let $l$ be a completely regular frame and $mathcal{r}l$ be the ‎ring of continuous real-valued functions on $l$‎. ‎we show that the‎ ‎lattice $zid(mathcal{r}l)$ of $z$-ideals of $mathcal{r}l$ is a‎ ‎normal coherent yosida frame‎, ‎which extends the corresponding $c(x)$‎ ‎result of mart'{i}nez and zenk‎. ‎this we do by exhibiting‎ ‎$zid(mathcal{r}l)$ as a quotient of $rad(mathcal{r}l)$‎, ‎the‎ ‎...

known as the left big finitistic projective dimension of A, is finite. Here pdM denotes the projective dimension of M . Unfortunately, this number is not known to be finite even if A is a finite dimensional algebra over a field, where, indeed, its finiteness is a celebrated conjecture. On the other hand, for such an algebra, finite flat certainly implies finite projective dimension, simply beca...

We introduce the notion of a projectively simple ring, which is an infinite-dimensional N-graded k-algebra A such that dimk A/I < ∞ for every nonzero 2-sided ideal I ⊂ A. We show that if a projectively simple ring A is strongly noetherian, is generated in degree 1, and has a point module, then A is equal in large degree to a twisted homogeneous coordinate ring B = B(X,L, σ). Here X is a smooth ...

An infinite-dimensional N-graded k-algebra A is called projectively simple if dimk A/I < ∞ for every nonzero two-sided ideal I ⊂ A. We show that if a projectively simple ring A is strongly noetherian, is generated in degree 1, and has a point module, then A is equal in large degree to a twisted homogeneous coordinate ring B = B(X,L, σ). Here X is a smooth projective variety, σ is an automorphis...