UDC 512.64, 535.1, 535-4 Degenerate 4-Dimensional Matrices with Semi-Group Structure and Polarization Optics E. M. Ovsiyuk∗, V. M. Red’kov† ∗ Mozyr State Pedagogical University, Belarus 28 Studencheskaya Street 247760, Mozyr, Gomel region Belarus † Institute of Physics, NAS of Belarus 220072 Minsk, Republic of Belarus, Nezavisimosty avenue, 68 In polarization optics, an important role play Mueller matrices — real four-dimensional matrices which describe the effect of action of optical elements on the polarization state of the light, described by 4-dimensional Stokes vectors. <...> An important issue is to classify possible classes of the Mueller matrices. <...> In particular, of special interest are degenerate Mueller matrices with vanishing determinants. <...> With the use of a special technique of parameterizing arbitrary 4-dimensional matrices in Dirac basis, a classification of degenerate 4-dimensional real matrices of rank 1, 2, 3. is elaborated. <...> To separate possible classes of degenerate matrices we impose linear restrictions on 16 parameters of 4 Ч 4 matrices which are compatible with the group multiplication law. <...> In polarization optics, an important issue is to classify possible classes of the Mueller matrices — an extensive list of references on the subject is given in [1]. <...> In particular, of special interest are degenerate Mueller matrices with vanishing determinants. <...> There is known a special technique of parameterizing arbitrary 4-dimensional matrices with the use of four 4-dimensional vector (k,m,l,n) — see [2, 3] and references therein. <...> To separate possible simple classes of degenerate matrices of ranks 1, 2 and 3 we impose linear restrictions on (k,m,l,n), which are compatible with the group multiplication law. <...> To obtain singular matrices of rank 3, we specify 16 independent possibilities to get the 4-dimensional matrices with zero determinant. <...> First, consider the variants with one independent vector. <...> We are to find all solutions of Eqs. (25); each of them will represent a sub-group or semi-group. <...> Here there are only 7 types of solutions, among 7 types of solutions, 6 cases lead to the structure of semigroup (matrices with rank 2). <...> We now consider the cases of two independent vectors. <...> The first solution <...>