Certain trace inequalities for positive definite matrices are generalized for positive semidefinite matrices using the notion of the group generalized inverse.

In this paper we develop a variety theory for unranked tree languages and unranked algebras. In an unranked tree any symbol may label a node with any number of successors. Such trees appear in markup languages such as XML and as syntactic descriptions of natural languages. In the corresponding algebras each operation is defined for any number of arguments, but in the regular algebras used as tree recognizers the operations are finite-state computable. We develop the basic theory of regular algebras for a setting in which algebras over different operator alphabets are considered together. Using syntactic algebras of unranked tree languages we establish a bijection between varieties of unranked tree languages and varieties of regular algebras. As varieties of unranked tree languages are usually defined by means of congruences of term algebras, we introduce also varieties of congruences and a general device for defining such varieties. Finally, we show that the natural unranked counterparts of several varieties of ranked tree languages form varieties in our sense.

In [5] M. Biliotti, V. Jha and N. Johnson were able to completely determine the autotopism group of a generalized twisted field as a subgroup of L(K) L(K), where K = GF (p n ) and L(K) is the group of nonsingular semilinear transformations over K. In this article, we consider the Cordero-Figueroa semifield of order 3 6 , which is not a generalized twisted field, and we prove that its autotopism group is isomorphic to a subgroup of L(K) L(K), where K = GF (3 6 ).

A non-deterministic hypersubstitution maps operation symbols to sets of terms of the corresponding arity.A non-deterministic hypersubstitution of type τ is said to be linear if it maps any operation symbol to a set of linear terms of the corresponding arity.We show that the extension of non-deterministic linear hypersubstitutions of type τ map sets of linear terms to sets of linear terms.As a consequence, the collection of all nondeterministic linear hypersubstitutions forms a monoid.Non-deterministic linear hypersubstitutions can be applied to identities and to algebras of type τ .

We introduce the concept of very true operator on a commutative basic algebra in a way analogous to that for fuzzy logics. We are motivated by the fact that commutative basic algebras form an algebraic axiomatization of certain non-associative fuzzy logics. We prove that every such operator is fully determined by a certain relatively complete sublattice provided its idempotency is assumed.

In this paper we show that there exists an infinite family of pairwise non-isomorphic entropic quasigroups with quasi-identity which are directly indecomposable and they are two-generated.