An existence variety (or e-variety) of regular semigroups is a class of regular semigroups which is closed under [Formula: see text], and ℍ. This concept was introduced by T.E. Hall and independently for orthodox semigroups by J. Kadourek and M.B. Szendrei who called them bivarieties. In this paper we prove the existence of e-free objects in each e-variety of E-solid regular semigroups and in each e-variety of locally inverse regular semigroups. By contrast, we show that there is no e-free object in other e-varieties.

The aim of this paper is to investigate homomorphisms which preserve square-free languages or primitive languages. A characterization of square-free-preserving homomorphisms is presented. We show that every square-free-preserving homomorphism is primitive-preserving. Strongly cube-free-preserving homomorphisms axe also studied

Suppose G is a simple graph with n vertices, m edges, and rank r. Let χGptq " a0t n ´a1t n´1 `¨¨¨`p´1q r art n´r be the chromatic polynomial of G. For q, k P Z and 0 ď k ď q `r `1, we obtain a sharp two-side bound for the partial binomial sum of the coefficient sequence, that is,Indeed, this bound holds for the characteristic polynomial of hyperplane arrangements and matroids, and its weak version can be generalized to the characteristic polynomial of toric arrangements and arithmetic matroids. We also propose a problem on the geometric interpretation of the above bound.

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Many systems involve substitutions between some sets of elements. The 0L system is a known technique which can help us to investigate properties of substitutions systematically. The aim of this paper is to establish some properties of the P0L schemes which preserve some types of properties of languages. Characterizations of pure-language-preserving, dense-preserving and palindrome-preserving P0L schemes are proposed. s-Injective, primitivity preserving, d-primitivity preserving, prefix code preserving and maximal prefix code preserving substitutions are also studied. Properties of dense-generating 0L schemes are also investigated