Matrix identities involving multiplication and transposition

Abstract

Abstract. We study matrix identities involving multiplication and unary operations such as transposition or Moore-Penrose inversion. We prove that in many cases such identities admit no finite basis. Background and motivation

The Finite Basis Problem for Kiselman Monoids

“…Since a 1 occurs inv 2 and, in view of (7), a 3 occurs inv [2,3] 1 , the leftmost occurrence of a 3 precedes the rightmost occurrence of a 1 in b. We see that 3 is either 1 or a power of a 2 , see Fig. 1, in which s stands for the element obtained fromv [2,3] 1 by removing all occurrences of a 3 except the leftmost one, while t denotes the element obtained fromv 2 by removing all occurrences of a 1 except the rightmost one and all occurrences of a 3 .…”

“…Hence we may assume that ℓ = 2 and a 3 ∈ c(x). Then 3] =x [3,3] must be a power of a 3 and, since a 2 3 = a 3 , we conclude that b =v [2,3] 1v 2 , d = a 3v [2,3] 1v 2 = a 3 b. Clearly, Lemma 4 allows us to retain in b only the leftmost occurrence of a 3 and only the rightmost occurrence of a 1 .…”

“…Applying Lemma 4a as in the proof of Theorem 8, we can rewrite b and d as follows: [2,3] ℓ−1v ℓ . If ℓ > 2, thenx [ℓ+1,3] = 1 and b = d. The same conclusions follow provided that a 3 / ∈ c(x).…”

“…If b 1 = 1 while b 2 = 1, we observe that b 3 = 1 since a 2 occurs inv 2 and thus in t. Therefore, b has the product a 2 a 3 a 1 a 2 as a factor. Both a 3 a 2 a 1 and a 2 a 3 a 1 a 2 belong to the ideal K whence in either of the two remaining cases we have b ∈ K and d = a 3 b ∈ K. · · ·v [2,3] ℓ−1v ℓ . If k + ℓ > 2, thenx [ℓ+1,3−k] = 1 and b = d. The same conclusions follow if a 2 / ∈ c(x).…”

“…The finite basis problem for involuted Kiselman monoids. A semigroup S is called an involuted semigroup if it admits a unary operation s → s * (called involution) such that (st) * = t * s * and (s * ) * = s for all s, t ∈ S. Whenever S is equipped with a natural involution, it appears to be reasonable to investigate the identities of S as an algebra of type (2,1); see [2,3,4,17] for numerous examples of recent studies along this line. Observe that adding an involution may radically change the equational properties of a semigroup: a finitely based semigroup may become a nonfinitely based involuted semigroup and vice versa.…”