Abstract
Abstract. Let the sign components of the maximal covectors of a simple oriented matroid M be represented by the real numbers −1 and 1. Consider the vertex set V(R) of a symmetric cycle R of adjacent topes in the tope graph of M as a subposet of the tope poset of M. If B is the bottom element of the tope poset then B is equal to the unweighted sum of the members of the set min V(R) of minimal elements of the subposet V(R); if B is the positive tope then the set min V(R) is a critical tope committee for the acyclic oriented matroid M.
Introduction
Let M := (E t , T ) be an oriented matroid, of rank ≥ 2, on the ground set E t := {1, . . . , t}, with set of topes T ; throughout we will suppose that it is simple, that is, it contains no loops, parallel or antiparallel elements. In this paper the sign components − and + of maximal covectors are replaced by the real numbers −1 and 1, respectively. The topes T := (T (1), . . . , T (t)) ∈ T are interpreted as elements of the real Euclidean space R t of row vectors; if T, T , T ∈ R t then T , T := t e=1 T (e) · T (e), and T := T, T . We denote by σ(1), . . . , σ(t) the standard basis of R t , that is, σ(i) := (0, . . . , 1 ↑ i , . . . , 0), 1 ≤ i ≤ t. The positive tope T (+) is the vector (1, . . . , 1) = t i=1 σ(i). If e ∈ E t then the corresponding positive halfspace T + e is the tope subset {T ∈ T : T (e) = 1}.
Introduction
If T ∈ T and A ⊆ E t then −A T denotes the vector obtained from T by sign reversal or reorientation on the set A: ( −A T )(e) = −T (e) when e ∈ A, Key words and phrases. Critical committee, oriented matroid, tope, tope graph, tope poset.
Introduction
2010 Mathematics Subject Classification: 05E45, 52C40, 90C27.
Introduction
and ( −A T )(e) = T (e) when e ∈ A. The oriented matroid whose topes are obtained from the topes of M by reorientation on the set A is denoted by −A M. We denote by T − the negative part {e ∈ E t : T (e) = −1} of the tope T ; the positive part T + of T is the set {e ∈ E t : T (e) = 1}.
Introduction
The vertices of the tope graph T (L(M)) of the oriented matroid M are its topes; a pair of topes {T , T } ⊂ T is an edge of the graph T (L(M)) if the topes T and T are adjacent, that is, they cover some subtope in the big face lattice of M. The separation set S(T , T ) of topes T and T is defined by S(T , T ) := {e ∈ E t : T (e) = T (e)}. The graph distance d(T , T ) between the topes T and T is the cardinality of the separation set S(T , T ), see [1, Prop. 4.2.3] , that is,
Introduction
If B ∈ T then we denote by T (L(M), B) the tope poset of M based at the tope B; by convention, we have 
Introduction
Let (l 1 , . . . , l t ) ∈ N t be the sequence defined by
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The chain m − {−B} is a basis of the space R t ; indeed, the square sign matrix
Introduction
is similar to the nonsingular matrix
Introduction
. . .
Introduction
the absolute value of its determinant is 2 t−1 .
Introduction
The
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Thus, if T ∈ T then we have T = xM for some row vector
Introduction
see [2, 3, 4, 5, 6] . The committee K * is called minimal if any its proper subset is not a committee for M. If the sum of the members of the minimal tope committee K * is the positive tope,
Introduction
then we say that the committee K * is critical .
Introduction
Recall that if R is a symmetric cycle in the tope graph T (L(M)) of the oriented matroid M, then for any tope T ∈ T there exists a unique inclusion-minimal subset
Introduction
this set, of odd cardinality, of linearly independent elements of R t is
Introduction
where the vector x is defined by (1.1).
Introduction
As a consequence, a tope subset K * ⊂ T is a committee for M iff
Introduction
In Section 2 we discuss relation (1.2) and describe the structure of the sets Q(T, R) in more detail.
Topes and Critical Committees
For a tope T of the oriented matroid M and for a symmetric cycle R in its tope graph, we describe the corresponding inclusion-minimal subset Q(T, R) ⊂ V(R) from expression (1. 
Topes and Critical Committees
is the inclusion-minimal subset of topes with the property T ∈max + ( −(B − ) V(R)) T = T (+) , that is,
Topes and Critical Committees
This means that min −(B − ) V(R) is a critical tope committee for the acyclic oriented matroid −(B − ) M. Relation (2.1) is equivalent to (2.2).
Topes and Critical Committees
Corollary 2.2. If R is a symmetric cycle in the tope graph of the oriented matroid M, then for any tope T ∈ T the corresponding set Q(T, R) is the set
Topes and Critical Committees
Example 2.3. Consider the Hasse diagram, which is depicted in Figure 1 Figure 1(b) . We have min −(B − ) V(R) = (−1, −1, 1, 1, 1), (1, 1, −1, −1, 1) , -1,1,1,1)  (-1,1,-1,1,1)  (-1,1,1,1,-1)   (1,-1,1,1,1)  (-1,1,-1,-1,1)  (-1,-1,1,1,-1 ) (-1,1,-1,1,-1) (1, -1,1,-1,1) (1,1,-1,-1,1) (1,-1,1,1,-1) (-1,1,-1,-1,-1) (1, -1,-1,-1,1)  (1,-1,1,-1,-1)  (1,1,-1,-1,-1) (1,-1,-1,-1,-1)
Topes and Critical Committees
(1,-1,1,1,1) (1,1,-1,1,1)  (1,1,1,1,-1)   (-1,-1,1,1,1) (1,1,-1,-1,1) (1,-1,1,1,-1) (1,1,-1,1,-1)   (-1,-1,1,-1,1) (-1,1,-1,-1,1)   (-1,-1,1,1,-1)  (1,1,-1,-1,-1)   (-1,-1,-1,-1,1)  (-1,-1,1,-1,-1)  (-1,1,-1,-1,-1 ) 1, 1, 1, 1 ) .
Topes and Critical Committees
Let R := (−1, 1, 1, 1, 1), (−1, −1, 1, 1, 1), (1, −1, 1, 1, 1), (1, −1, 1, −1, 1 (-1,-1,1,1,1 ) (-1,1,-1,1,1)  (-1,1,1,1,-1 ) -1,1,-1,1) (1,1,-1,-1,1)  (1,-1,1,1,-1)  (-1,1,-1,-1,-1) (1,-1,-1,-1,1) (1,-1,1,-1,-1)  (1,1,-1,-1,-1) (1,-1,-1,-1,-1) 1, 1, 1, 1) , and the subposet V(R) which is the vertex set of a symmetric cycle R in the tope graph, cf. Figures 1(a) and 2 . (−1, 1, −1, 1, −1), (−1, 1, 1, 1, −1), (−1, 1, 1, 1, 1 ) be one more symmetric cycle in the tope graph of the oriented matroid M, see 

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