User:Rschwieb/Cold storage

• • Steiner magma: A commutative magma satisfying x.xy = y.
    • Squag: an idempotent Steiner magma.^[1]
    • Sloop: a Steiner magma with distinguished element 1, such that xx = 1.
  • Equivalential algebra: a magma satisfying xx.y=y, xy.z.z=xy, and xy.xzz.xzz=xy.^[2]
  • Implicational calculus: a magma satisfying xy.x=x, x.yz=y.xz, and xy.y=yx.x.^[3]
  • Equivalence algebra: an idempotent magma satisfying xy.x=x, x.yz=xy.xz, and xy.z.y.x = xz.y.x.^[4]
  • Semigroup: an associative magma.
    • Equivalential calculus: a commutative semigroup satisfying yyx=x.^[5]
      • Boolean group: a monoid with xx = identity element.
        • • Reductive group: an algebraic group such that the unipotent radical of the identity component of S is trivial.
      • Logic algebra: a commutative monoid with a unary operation, complementation, denoted by enclosure in parentheses, and satisfying x(1)=(1) and ((x))=x. 1 and (1) are lattice bounds for S.
        • MV-algebra: a logic algebra satisfying the axiom ((x)y)y = ((y)x)x.
        • Boundary algebra: a logic algebra satisfying (x)x=1 and (xy)y = (x)y, from which it can be proved that boundary algebra is a distributive lattice. (0)=1, (1)=0, ((x))=x and xx=x are now provable.
  • Order (algebra): an idempotent magma satisfying yx=xy.x, xy=xy.y, x:xy.z=x.yz, and xy.z.y=xz.y. Hence idempotence holds in the following wide sense. For any subformula x of formula z: (i) all but one instance of x may be erased; (ii) x may be duplicated at will anywhere in z.
    • Band: an associative order algebra, and an idempotent semigroup.
      • Rectangular band: a band satisfying the axiom xyz = xz.
      • Normal band: a band satisfying the axiom xyzx = xzyx.

The following two structures form a bridge connecting magmas and lattices:

• • Newman algebra: a ringoid that is also a shell. There is a unary operation, inverse, denoted by a postfix "'", such that x+x'=1 and xx'=0. The following are provable: inverse is unique, x"=x, addition commutes and associates, and multiplication commutes and is idempotent.
  • Semiring: a ringoid that is also a shell. Addition and multiplication associate, addition commutes.
    • Commutative semiring: a semiring whose multiplication commutes.
  • Rng: a ringoid that is an Abelian group under addition and 0, and a semigroup under multiplication.
    • Ring: a rng that is a monoid under multiplication and 1.
      • Commutative ring: a ring with commutative multiplication.
        • Boolean ring: a commutative ring with idempotent multiplication, isomorphic to Boolean algebra.
      • Differential ring: A ring with an added unary operation, derivation, denoted by prefix ∂ and satisfying the product rule, ∂(xy) = ∂xy+x∂y.
  • Bounded lattice: a lattice with two distinguished elements, the greatest (1) and the least element (0), such that x∨1=1 and x∨0=x.
  • Involutive lattice: a lattice with a unary operation, denoted by postfix ', and satisfying x"=x and (x∨y)' = x' ∧y' .
  • Relatively complemented lattice:
  • Complemented lattice: a lattice with 0 and 1 such that for any x there is y with x ∨ y = 1 and x∧y = 0. Not definable by identities
  • Lattice with choice of complement: a lattice with a unary operation, [complementation]], denoted by postfix ', such that x∧x' = 0 and 1=0'. 0 and 1 bound S -- as well as the dual conditions.
    • Orthocomplemented lattice: a lattice with complementation satisfying x" = x and x∨y=y ↔ y' ∨x' = x' (complementation is order reversing).
      • Orthomodular lattice: an ortholattice such that (x ≤ y) → (x ∨ (x^⊥ ∧ y) = y) holds.
    • De Morgan algebra: a complemented lattice satisfying x" = x and (x∨y)' = x' ∧y' . Also a bounded involutive lattice.
  • Modular lattice: a lattice satisfying the modular identity, x∨(y∧(x∨z)) = (x∨y)∧(x∨z).
    • Metric lattice: not definable by identities
    • Projective lattice: not definable by identities
    • Arguesian lattice: a modular lattice satisfying the identity
    • Distributive lattice: a lattice in which each of meet and join distributes over the other. Distributive lattices are modular, but the converse need not hold.
      • Boolean algebra: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation.
        • Boolean algebra with operators: a Boolean algebra with one or more added operations, usually unary. Let a postfix * denote any added unary operation. Then 0* = 0 and (x∨y)* = x*∨y*. More generally, all added operations (a) evaluate to 0 if any argument is 0, and (b) are join preserving, i.e., distribute over join.
          • Modal algebra: a Boolean algebra with a single added operator, the modal operator.
            • Derivative algebra: a modal algebra whose added unary operation, the derivative operator, satisfies x**∨x*∨x = x*∨x.
            • Interior algebra: a modal algebra whose added unary operation, the interior operator, satisfies x*∨x = x and x** = x*. The dual is a closure algebra.
              • Monadic Boolean algebra: a closure algebra whose added unary operation, the existential quantifier, denoted by prefix ∃, satisfies the axiom ∃(∃x)' = (∃x)'. The dual operator, ∀x := (∃x' )' is the universal quantifier.

Three structures whose intended interpretations are first order logic:

• Polyadic algebra: a monadic Boolean algebra with a second unary operation, denoted by prefixed S. I is an index set, J,K⊂I. ∃ maps each J into the quantifier ∃(J). S maps I→I transformations into Boolean endomorphisms on S. σ, τ range over possible transformations; δ is the identity transformation. The axioms are: ∃(∅)a=a, ∃(J∪K) = ∃(J)∃(K), S(δ)a = a, S(σ)S(τ) = S(στ), S(σ)∃(J) = S(τ)∃(J) (∀i∈I-J, such that σi=τi), and ∃(J)S(τ) = S(τ)∃(τ^-1J) (τ injective).^[6]
• Relation algebra: S, the Cartesian square of some set, is a:

• Boolean algebra under join and complementation;
• Monoid under binary composition (infix •) and the identity element I such that 1=I '∨I;
• Residuated Boolean algebra by virtue of a second unary operation, converse (postfix ^{${\displaystyle {\breve {\ }}}$}) and the axiom (A^{${\displaystyle {\breve {\ }}}$}•(A•B)')∨B ' = B '.

Converse is an involution and distributes over composition so that (A•B)^{${\displaystyle {\breve {\ }}}$} = B^{${\displaystyle {\breve {\ }}}$}•A^{${\displaystyle {\breve {\ }}}$}. Converse and composition each distribute over join.^[7]

• Cylindric algebra: Boolean algebra augmented by unary cylindrification operations.

Others:

• • • • • Coalgebra: the dual of a unital associative algebra.
        • Incidence algebra: an associative algebra such that the elements of S are the functions f [a,b]: [a,b]→R, where [a,b] is an arbitrary closed interval of a locally finite poset. Vector multiplication is defined as a convolution of functions.
        • Kac-Moody algebra: a Lie algebra, usually infinite-dimensional, definable by generators and relations through a generalized Cartan matrix.
          • Generalized Kac-Moody algebra: a Kac-Moody algebra whose simple roots may be imaginary.
          • Affine Lie algebra: a Kac-Moody algebra whose generalized Cartan matrix is positive semi-definite and has corank 1.