Semigroup Forum Vol. 61 (2000) 363–372 c  2000 Springer-Verlag New York Inc. RESEARCH ARTICLE The Five-Element Brandt Semigroup as a Forbidden Divisor ∗ Miroslav ´ Ciri´ c and Stojan Bogdanovi´ c Communicated by M. S. Putcha Abstract The purpose of this paper is to describe all semigroups which do not have the five-element Brandt semigroup B 2 among its divisors. These semigroups will be described in two ways: as semigroups which have certain property as a hereditary property, and as semigroups which satisfy some variable identities. We also describe all variable identities over the two-element alphabet that determine variable varieties contained in the considered class of semigroups. 1. Introduction and preliminaries The five-element Brandt semigroup B 2 was studied many times as a forbidden member of certain classes of semigroups. For example, pseudovarieties which do not contain B 2 were described by S. W. Margolis in [11] (see also M. S. Putcha [16] and J. E. Pin [14] and [15]). Varieties which do not contain B 2 were completely characterized by the authors in [6] and [7], and in the periodic case by M. V. Sapir and E. V. Suhanov in [19]. Epigroups (semigroups equipped with one special unary operation) which do not have B 2 among its divisors were investigated by L. N. Shevrin in [20]. The purpose of this paper is to describe all semigroups having B 2 as a forbidden divisor. We do it in two ways. First, we describe these semigroups as semigroups which have one particular property, studied previously by the authors in [3], as a hereditary property. This method was used in many papers, and particularly, it was one of the central methods used in the book [2] of the second author. On the other hand, the class of all semigroups having B 2 as a forbidden divisor will be characterized in terms of variable identities, namely as a variable variety. This is the same concept which was introduced by M. S. Putcha and J. Weissglass in [17] and [18], but we give another definition that is closer to the definition of ordinary identities and varieties than one of Putcha and Weissglass. In some sense, this concept traces one’s origin to the concept of pseudo identities and pseudo varieties, introduced by B. M. Schein in 1960s (or disjunctive identities and varieties, as they were called in [12]). These are universal formulas given as disjunctions of equalities 2 . The related concepts, so-called inclusive identities and collective identities, were studied in [1], [9], [10] and [12]. * Supported by Grant 04M03B of RFNS through Math. Inst. SANU. 2 One example is the following universal formula: xyz = xy ∨ xyz = yz ∨ xyz = xz . Semigroups satisfying this formula were called exclusive, and their investigation was also initiated by B. M. Schein. They were studied in [13], [21] and [22].