The P,-reducible graphs are a natural generalization of the well-known class of cographs, with applications to scheduling, computational semantics, and clustering. More precisely, the Pa-reducible graphs are exactly the graphs none of whose vertices belong to more than one chordless path with three edges. A remarkable property of P,-reducible graphs is their unique tree representation up to iso...

An n-ary operation Q : Σ → Σ is called an n-ary quasigroup of order |Σ| if in the equation x0 = Q(x1, . . . , xn) knowledge of any n elements of x0, . . . , xn uniquely specifies the remaining one. Q is permutably reducible if Q(x1, . . . , xn) = P ( R(xσ(1), . . . , xσ(k)), xσ(k+1), . . . , xσ(n) ) where P and R are (n− k+1)-ary and kary quasigroups, σ is a permutation, and 1 < k < n. Anm-ary ...

It is shown that, for every k>0 and every fixed algorithmically random language B, there is a language that is polynomial-time, truth-table reducible in k+1 queries to B but not truth-table reducible in k queries in *any* amount of time to *any* algorithmically random language C. In aprticular, this yields the separation P (RAND) is a proper subset of P (RAND), k-tt (k+1)-tt where RAND is the s...

Let $M$ be a module over a commutative ring $R$ and let $N$ be a proper submodule of $M$. The total graph of $M$ over $R$ with respect to $N$, denoted by $T(Gamma_{N}(M))$, have been introduced and studied in [2]. In this paper, A generalization of the total graph $T(Gamma_{N}(M))$, denoted by $T(Gamma_{N,I}(M))$ is presented, where $I$ is an ideal of $R$. It is the graph with all elements of $...

The graded-fermion algebra and quasi-spin formalism are introduced and applied to obtain the gl(m|n) ↓ osp(m|n) branching rules for the “two-column” tensor irreducible representations of gl(m|n), for the case m ≤ n (n > 2). In the case m < n, all such irreducible representations of gl(m|n) are shown to be completely reducible as representations of osp(m|n). This is also shown to be true for the...

Let G be a connected reductive linear algebraic group. We use geometric methods to investigate G-completely reducible subgroups of G, giving new criteria for G-complete reducibility. We show that a subgroup of G is G-completely reducible if and only if it is strongly reductive in G; this allows us to use ideas of R.W. Richardson and Hilbert– Mumford–Kempf from geometric invariant theory. We ded...