L.A

1.

LATTICES AND HYPERIDENTITIES IN THE ALGEBRAIC MODELING OF ARCHITECTURAL SYSTEMS
YURI MOVSISYAN, YSU
LIANA ABRAHAMYAN, NUACA,EIU

2. Lattices and hyperidentities in the algebraic modeling of architectural systems

We propose an algebraic framework for modeling
architectural systems using lattices. Spaces, structural
elements and their dependencies form partially ordered
structures, while hyperidentities express design rules that
remain valid under any substitution of operations. This
guarantees consistency of rules across design models and
enables systematic verification of architectural
configurations.

3. Why formal models in architecture?

• A building combines interdependent spaces,
structure, envelope and services
• Informal design rules can contradict one
another
• A formal language can check consistency
early

4. Lattices: ordered structure

• A partially ordered set where any two
elements have a join ∨ and a meet ∧
• Models hierarchies of spaces and building
subsystems
• Obeys commutative, associative and
absorption laws

5. Hyperidentities

• An identity holds for all elements
• A hyperidentity holds for all substitutions of
operation symbols by term operations
(Taylor, 1981)
• It states a rule independent of any concrete
operation

6. Aim and tasks

• Aim: a lattice framework for architectural
systems
• Task 1: find hyperidentities valid in all
lattices
• Task 2: explain their meaning for
architectural design rules

7. Modeling scheme

Architecture
Lattice
Space or element
element
Is part of
order ≤
Union of zones
join ∨
Shared area
meet ∧

8. Algebraic setting

• Algebra (L, ∧, ∨) from the variety of lattices
• Binary term operations: x, y, x ∧ y, x ∨ y
• Symbols F and G stand for these four

9. Testing a hyperidentity

1
Replace each
operation symbol
by every binary
term operation
2
Replace each
operation symbol
by every ternary
term operation
3
Check that the law
holds in every case

10. Reading the outcome

• A hyperidentity is a design rule that holds
however zones are combined
• A failed law depends on the chosen
operation

11. Four substitutions suffice

x
y
x∧y
x∨y
Every binary term operation of a lattice is one of these four, so
the check is finite

12. Hyperidentities of lattices

Each holds for F = x, y, x ∧ y, x ∨ y
Hyperidempotency
F(x, x) = x
Hyperassociativity
F(x, F(y, z)) = F(F(x, y), z)
Hypermediality
F(F(x, y), F(z, t)) = F(F(x, z), F(y, t))

13. Laws that fail

Commutativity
Absorption
F(x, y) = F(y, x)
F(x, G(x, y)) = x
Fails for F(x, y) = y
Fails for F = y, G = x ∨ y
These laws depend on the chosen operation

14. Example: a building as a lattice

SEM
S structure
SE
SM
EM
E envelope
M services
S
E
∅
M
SE = S ∨ E
∅ empty, SEM whole
building

15. Meaning for architectural design

Valid whatever combination rule is used
1
Idempotency: a repeated space adds nothing
2
Associativity: the order of merging wings is irrelevant
3
Mediality: a 2 × 2 module grid can be assembled by rows or by
columns

16. Discussion

• Rules that survive every substitution are
structural, hence safe to reuse
• Other rules are tied to a particular operation
• Results are conceptual; validation on real
architectural projects is needed

17. Conclusions

• Lattices model spatial and structural
hierarchies
• Hyperidentities separate structural rules
from operation-specific ones
• Three hyperidentities hold in all lattices

18. Outlook

• Extend to Boolean algebras and other
varieties
• Software tool to verify design rules, e.g. in
BIM
• Case studies on real buildings

19. References (1/2)

1. Birkhoff G. Lattice Theory, 3rd ed. AMS, 1967.
2. Grätzer G. General Lattice Theory, 2nd ed. Birkhäuser, 2003.
3. Davey B.A., Priestley H.A. Introduction to Lattices and Order, 2nd ed.
Cambridge Univ. Press, 2002.
4. Burris S., Sankappanavar H.P. A Course in Universal Algebra. Springer, 1981.

20. References (2/2)

5. Taylor W. Hyperidentities and hypervarieties. Aequationes Math. 23 (1981)
30–49.
6. Movsisyan Yu.M. Hyperidentities in algebras and varieties. Russian Math.
Surveys 53 (1998) 57–108.
7. Movsisyan Yu.M. Introduction to the Theory of Algebras with Hyperidentities.
Yerevan State Univ. Press, 1986.
8. Denecke K., Wismath S.L. Universal Algebra and Applications in Theoretical
Computer Science. Chapman & Hall/CRC, 2002.