G. Gratzer

collaborators

4 papers

math.RA2025

Notes on the ordered set $A^{A}$ . Part IV. The dual $^{A} A$ of $A^{A}$ for finite ordered sets

G. GrÃ¤tzer

Let $A$ be a finite ordered set. Define the ordered set $A^{A}$ as the set of all maps from $A$ to $A$ , ordered pointwise. Let $^{A} A$ be the dual of $A^{A}$ . We prove results in th…

math.RA2025

Notes on the ordered set $A^{A}$ . Part III. Exponent-cancellations

G. GrÃ¤tzer

We continue the study of exponent-cancellation for finite ordered sets. It is known that $A$ can be reconstructed from $A^{A}$ , from $(A^{A})^{A}$ , and from $A^{A^{A}}$ . In this no…

math.CO2025

Notes on the ordered set $A^{A}$ II. Higher Exponentials

George GrÃ¤tzer

For the finite ordered sets $A, D$ , write $A^{D}$ for the ordered set of isotone maps $D \to A$ with the pointwise order. It was proved in earlier work that the order structure of $A…

math.RA2025

Notes on the ordered set $A^{A}$ . Part I. The classical problem

G. GrÃ¤tzer

Let $A$ and $B$ be finite ordered sets. We show that if the ordered sets of isotone self-maps $A^{A}$ and $B^{B}$ (ordered pointwise) are isomorphic, then $A$ and $B$ are isomorphic. T…

Notes on the ordered set AA. Part IV. The dual AA of AA for finite ordered sets

Notes on the ordered set AA. Part III. Exponent-cancellations

Notes on the ordered set AA II. Higher Exponentials

Notes on the ordered set AA. Part I. The classical problem

Notes on the ordered set $A^{A}$ . Part IV. The dual $^{A} A$ of $A^{A}$ for finite ordered sets

Notes on the ordered set $A^{A}$ . Part III. Exponent-cancellations

Notes on the ordered set $A^{A}$ II. Higher Exponentials

Notes on the ordered set $A^{A}$ . Part I. The classical problem