Ternary Term Research Articles

Decidable varieties with modular congruence lattices

For a large collection of varieties we show that if the first-order theory of such a variety is decidable then the variety decomposes into the product of two well-known highly specialized varieties. For many varieties the decidability question then reduces to a decidability question about modules. A variety is a class of (abstract) algebras (belonging to some language) closed under the formation of direct products, subalgebras and homomorphic images. A variety 1/ is locally finite if every finitely-generated member of (/ is finite. A class of algebras K generates a variety 1/ if 1/ is the smallest variety containing K—then we say that \J is generated by K, written 1/ = V{K). A variety is finitely generated if it is generated by finitely many finite algebras, or equivalently by a single finite algebra. The kernel of a homomorphism is called a congruence, and the congruences of any algebra form a lattice. A variety is congruence modular, or we prefer to say just modular, if the lattice of congruences of every algebra in the variety satisfies the modular law. (Most of the well-studied varieties are modular; for example varieties of groups, rings, modules and lattices are modular. However,the variety of semigroups is not modular.) A variety \] is a product of two subvarieties | / j , |/2 ifl / j U l/2 generates (/ and there is a term b(x, y) such that M x \= b{x, y) = x, l/2 |= b(x, y) = y. If this is so we write 1/ = 1/ x <£) 1/ 2 , and then for every algebra A in 1/ there are (up to isomorphism) unique algebras Ax G (/p A2 G (/2 such that A = Ax x A2. A class K of first-order structures has a decidable theory if there is an effective procedure to determine precisely which first-order sentences are true of every member of K. A variety 1/ is a discriminator variety if it is generated by a set K for which there exists a ternary term t(xf y, z) such that K satisfies t(x, yy z) = x if x =£ y = z if x — y. In everyday mathematics such varieties appear only as highly specialized varieties of rings, or varieties associated with algebraic logics. Received by the editors December 29, 1980. 1980 Mathematics Subject Classification. Primary 03B25, 08B10, 08A05.

Model analysis of mixing enthalpies in the Cd-TI-In ternary system

Three different solution models have been used to describe the experimental data for mixing enthalpies for liquid Cd-TI-In alloys. All models require that an additional ternary term of the form. z ABC - x A - x B - x c - be added to the summation of the binary mixing enthalpies of the constituent systems in order to obtain satisfactory agreement between experimental and calculated values.

Thermodynamics of ternary systems. Quasichemical approximation and the excess ternary term

Based on the quasichemical theory, second order expressions in terms of the nearest-neighbor interaction parameters are given for the molar energy of mixing, ΔE, and the partial molar excess free energy of mixing, ḠCE, of a ternary system. In order to improve the applicability, the quasichemical binary contributions are replaced by the experimental binary values. The ``leftover'' term or the ``excess'' ternary term is discussed in more detail. The quasichemical theory suggests the ternary excess energy term is of a more complex form than N AN BNC × const. The expressions are useful in calorimetric work and in discussing phase stabilities (miscibility gaps).

Thermodynamic modeling of binary and ternary metallic solutions

A model is proposed for describing heat of mixing behavior in binary and ternary metallic solutions. The binary model, which has the form, ΔH M =α1 X 2 X B +α2 X A X 2 −α3 X 2 X 2 , whereX A andX B are mole fractions of componentsA andB and α1, α2, and α3 are constants, is applied to the heat of mixing values for 84 solid and liquid systems and the results are compared with the subregular model. The ternary model, which is composed of the sum of the binary equations and a ternary interaction term of the form α ABC X A X B X C , was applied to the Bi−Cd−Pb, Cd−Pb−Sn, and Cd−Pb−Sb systems. There was excellent agreement both as to the shapes of the isoenthalpy of mixing curves and as to the heat of mixing values in the ternary systems when the model was used to predict the experimental values.