Abstract

The symmetric group $ S(n) $ is partially ordered by Bruhat order. This order is extended by L. Renner to the set of partial injective functions of $ \{ 1, 2, \ldots, n \} $ (see, Linear Algebraic Monoids, Springer, 2005). This poset is investigated by M. Fortin in his paper The MacNeille Completion of the Poset of Partial Injective Functions [Electron. J. Combin., 15, R62, 2008]. In this paper we show that Renner order can be also defined for sets of all functions, partial functions, injective and partial injective functions from $ \{ 1, 2, \ldots, n \} $ to $ \{ 1, 2, \ldots, m \} $. Next, we generalize Fortin's results on these posets, and also, using simple facts and methods of linear algebra, we give simpler and shorter proofs of some fundamental Fortin's results. We first show that these four posets can be order embedded in the set of $ n \times m $-matrices with non-negative integer entries and with the natural componentwise order. Second, matrix representations of the Dedekind-MacNeille completions of our posets are given. Third, we find join- and meet-irreducible elements for every finite sublattice of the lattice of all $ n \times m $-matrices with integer entries. In particular, we obtain join- and meet-irreducible elements of these Dedekind-MacNeille completions. Hence and by general results concerning Dedekind-MacNeille completions, join- and meet-irreducible elements of our four posets of functions are also found. Moreover, subposets induced by these irreducible elements are precisely described.

Highlights

  • The symmetric group S(n) of all bijections of the finite set {1, 2, . . . , n} is partially ordered by Bruhat order

  • In this paper we show that Renner order can be defined for sets of all functions, partial functions, injective and partial injective functions from {1, 2, . . . , n} to {1, 2, . . . , m}

  • The Bruhat order is extended by Renner in [12] for the set P (n) of all partial injective functions of {1, 2, . . . , n}

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Summary

Introduction

The symmetric group S(n) of all bijections of the finite set {1, 2, . . . , n} is partially ordered by Bruhat order (see [1]). The Dedekind-MacNeille completion of S(n) (i.e., the the electronic journal of combinatorics 23(1) (2016), #P1.3 smallest lattice that contains S(n)) is characterized (up to isomorphism) in [10] as some finite lattice of square matrices of size n with non-negative integer entries and with the natural componentwise order. It is obtained in [10] that this completion has the same number of elements as the set of all alternating sign matrices (see [2]). Structures of subposets induced by these irreducible elements are precisely described

Basic definitions and facts
Posets of functions
Lattices of matrices
Matrix representation
F G33 and
Dedekind-MacNeille completions
Basic notions and facts
B B f f l1 l2
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